Large groups of automorphisms of compact Riemann surfaces of genus 2 through 5
---------------------------------------------------------------------------

This is a truncation of a list published by Marston Conder (University of 
Auckland).  His list goes to genus 101 and was found with the help of the
"LowIndexNormalSubgroups" routine in Magma, in August 2010.  

Here "large" means the group G of orientation-preserving automorphisms 
has order |G| > 4(g-1), where g is the genus of the Riemann surface. 

This list is ordered by the *genus* of the Riemann surface.

In each case the group G is a quotient F/K, where K is a torsion-free normal
subgroup of the Fuchsian group F that defines the action, with triangular or 
quadrangular signature.  This subgroup K is the kernel of a smooth epimorphism 
from F to G, and has index |F:K| = |G| (the order of G).

An indication is given as to whether the action is reflexible, or chiral, 
and in the reflexible case, whether there's a corresponding action of G on a
compact non-orientable surface of (non-orientable) genus g+1.

The expressions in square brackets are defining relations for the group G
(in terms of three generators G.1, G.2 and G.3 in the triangular case, or four
generators G.1, G.2, G.3 and G.4 in the quadrangular case).  These are also
normal generators for the kernel K -- that is, K is their normal closure.

Following the group actions for each given genus, a note is made of 
(a) the largest group action preserving orientation,  
(b) the largest group action that would occur if we allowed orientation-reversing 
automorphisms as well, 
(c) the largest reflexible group action,  
(d) the largest chiral (irreflexible) group action, 
(e) the largest group action on a non-orientable surface of non-orientable 
genus g+1, together with the type(s) of the largest actions in each case. 


Genus 2 actions:

Genus 2 action of group of order 48 via triangle group [ 2, 3, 8 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.3 * G.2 * G.1 * G.3 * G.2^-1 * G.3^-1 * G.1 * G.3 ]

Genus 2 action of group of order 24 via triangle group [ 2, 4, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^6 ]

Genus 2 action of group of order 16 via triangle group [ 2, 4, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-1 * G.1 * G.2^-1 * G.3^-2 ]

Genus 2 action of group of order 10 via triangle group [ 2, 5, 10 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-5 ]

Genus 2 action of group of order 12 via triangle group [ 2, 6, 6 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^6 ]

Genus 2 action of group of order 8 via triangle group [ 2, 8, 8 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^3 * G.3^-1 ]

Genus 2 action of group of order 24 via triangle group [ 3, 3, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.3^4, G.3 * G.2 * G.1^-1 * G.2^-1 * G.1 ]

Genus 2 action of group of order 12 via triangle group [ 3, 4, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2 * G.1^-1 * G.3, G.2^4 ]

Genus 2 action of group of order 6 via triangle group [ 3, 6, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.2^-1, G.3 * G.2^-1 ]

Genus 2 action of group of order 8 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.2 * G.1^-1 * G.3, G.3^-1 * G.1^-1 * G.2 ]

Genus 2 action of group of order 5 via triangle group [ 5, 5, 5 ]
Action is reflexible
[ G.1^-2 * G.2^-1, G.2 * G.1^-1 * G.2, G.3 * G.1^-1 ]

Genus 2 action of group of order 5 via triangle group [ 5, 5, 5 ]
Action is reflexible
[ G.1^-2 * G.3^-1, G.3 * G.1^-1 * G.3, G.2 * G.1^-1 ]

Genus 2 action of group of order 5 via triangle group [ 5, 5, 5 ]
Action is reflexible
[ G.1^-2 * G.2, G.2^-1 * G.1^-1 * G.2^-1, G.3 * G.2^-1 ]

Genus 2 action of group of order 12 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4^-3 ]

Genus 2 action of group of order 12 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^-3 ]

Genus 2 action of group of order 12 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3 ]

Genus 2 action of group of order 8 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.4^-1 * G.1 * G.4^-1, (G.1 * G.2)^2, G.3 * G.2 * G.4 ]

Genus 2 action of group of order 8 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.1 * G.4, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1 ]

Genus 2 action of group of order 8 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.2 * G.1 * G.4, G.3 * G.2 * G.1 * G.4^-1 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, G.3^-1 * G.1 * G.3 * G.1, G.2 * G.1^-1 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, (G.3^-1 * G.1)^2, G.2 * G.1^-1 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.2 * G.1 * G.3, G.3^-1 * G.2 * G.1 * G.3^-1, G.4 * G.3^-1 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.2^-1, G.2^-3, G.2^-1 * G.1 * G.2 * G.1, G.3 * G.1^-1 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.4^-1 * G.2^-1, G.2^-3, G.1 * G.2^-1 * G.3 ]

Genus 2 action of group of order 6 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.2^-3, (G.2^-1 * G.1)^2, G.3 * G.1^-1, G.4 * G.2^-1 ]

Largest orientation-preserving group for genus 2 has order 48
 ... smooth quotient of triangle group [ 2, 3, 8 ]
Largest group with or without reflections for genus 2 has order 96
 ... smooth quotient of full triangle group [ 2, 3, 8 ]*
Largest group with reflections for genus 2 has order 96
 ... smooth quotient of full triangle group [ 2, 3, 8 ]*
Largest chiral group for genus 2 has order 6
 ... smooth quotient of quadrangle group [ 2, 3, 2, 3 ]
Largest non-orientable group for non-orientable genus 3 has order 12
 ... smooth quotient of full quadrangle group [ 2, 2, 2, 3 ]*
Check totals true : 25 : 2 chiral, 23 reflexible, plus 4 non-orientable


Genus 3 actions:

Genus 3 action of group of order 168 via triangle group [ 2, 3, 7 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.3^-3 * G.1 * G.2 * G.3^-3, G.2 * G.1 * G.3 * G.2 * G.1 * G.3 * G.2^-1 * G.3 * G.2^-1 * G.3^-1 * G.1 * G.3 ]

Genus 3 action of group of order 96 via triangle group [ 2, 3, 8 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.3^-4 * G.1 * G.2 * G.3^-3, G.3 * G.2 * G.1 * G.3 * G.2^-1 * G.3 * G.2^-1 * G.3^-1 * G.1 ]

Genus 3 action of group of order 48 via triangle group [ 2, 3, 12 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.1 * G.3^2 * G.2^-1 * G.3^-3 ]

Genus 3 action of group of order 48 via triangle group [ 2, 4, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-2)^2, G.3^6 ]

Genus 3 action of group of order 32 via triangle group [ 2, 4, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.1 * G.3 * G.2^-1 * G.3^-2 ]

Genus 3 action of group of order 32 via triangle group [ 2, 4, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-4 * G.1 * G.2 * G.3^-3 ]

Genus 3 action of group of order 24 via triangle group [ 2, 4, 12 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-2 * G.1 * G.2^-1 * G.3^-3 ]

Genus 3 action of group of order 24 via triangle group [ 2, 6, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.2^-2 * G.3^-2, G.2^6 ]

Genus 3 action of group of order 14 via triangle group [ 2, 7, 14 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-7 ]

Genus 3 action of group of order 16 via triangle group [ 2, 8, 8 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-8 ]

Genus 3 action of group of order 16 via triangle group [ 2, 8, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.3^-1 * G.2^2 * G.3^-1 ]

Genus 3 action of group of order 12 via triangle group [ 2, 12, 12 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^5 * G.3^-1 ]

Genus 3 action of group of order 48 via triangle group [ 3, 3, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.3^4, G.3 * G.1^-1 * G.2^-1 * G.1 * G.2^-1 * G.1 ]

Genus 3 action of group of order 24 via triangle group [ 3, 3, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.2^-1 * G.3^-1 * G.1 * G.3 ]

Genus 3 action of group of order 21 via triangle group [ 3, 3, 7 ]
Action is chiral
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.3 * G.2 * G.1 * G.3 ]

Genus 3 action of group of order 21 via triangle group [ 3, 3, 7 ]
Action is chiral
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.2^-1 * G.3^-1 * G.1^-1 * G.3 ]

Genus 3 action of group of order 24 via triangle group [ 3, 4, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.1^-1 * G.2^-1 * G.1 * G.3, G.2^4 ]

Genus 3 action of group of order 12 via triangle group [ 3, 4, 12 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.2^4 ]

Genus 3 action of group of order 9 via triangle group [ 3, 9, 9 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.2^-2 * G.3 ]

Genus 3 action of group of order 9 via triangle group [ 3, 9, 9 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.2^-2 * G.1 * G.2^-1 ]

Genus 3 action of group of order 16 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.1^4, G.2^4 ]

Genus 3 action of group of order 16 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.2 * G.1^-1 * G.3, G.1^4, G.2^4 ]

Genus 3 action of group of order 16 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3^4 ]

Genus 3 action of group of order 16 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2, G.1^4, G.2^4 ]

Genus 3 action of group of order 12 via triangle group [ 4, 4, 6 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3 * G.2^-1 * G.1 * G.3 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, G.4 * G.1 * G.3 * G.1 * G.4^-1 * G.2 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.3 * G.4^-1 * G.2 * G.4, G.4^-3 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.4^-3, G.4 * G.2 * G.1 * G.3 * G.1 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.3 * G.4^-1 * G.1 * G.4, G.4^-3 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, (G.3 * G.1)^3 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3, G.2 * G.4^-1 * G.2 * G.1 * G.3 * G.4^-1 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.4^-3, G.2 * G.3 * G.1 * G.3 * G.4^-1 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3, (G.2 * G.1)^3 ]

Genus 3 action of group of order 24 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.2 * G.4^-1 * G.1 * G.4, G.4^-3 ]

Genus 3 action of group of order 16 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^4 ]

Genus 3 action of group of order 16 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^4 ]

Genus 3 action of group of order 16 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4^4 ]

Genus 3 action of group of order 16 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.1 * G.3 * G.2 * G.4^-1 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4 * G.3 * G.2 * G.4 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4 * G.3 * G.1 * G.4 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.4 * G.2 * G.1 * G.4, (G.1 * G.3)^2 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, (G.1 * G.3^-1)^3, G.2 * G.1^-1 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3 * G.1 * G.4, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.2 * G.3^-1 * G.1 * G.3, G.3^-3 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.3 * G.2 * G.4, G.1 * G.3^-1 * G.1 * G.4^-1, G.3^-3 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.4^-1 * G.2^-1, G.2^-3, G.2^-1 * G.1 * G.2 * G.3, (G.1 * G.3)^2, G.2 * G.1 * G.2^-1 * G.1 * G.3 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.2^-3, G.2^-1 * G.1 * G.2^-1 * G.4, G.4^-1 * G.1 * G.2^-1 * G.1, G.3 * G.1^-1 ]

Genus 3 action of group of order 12 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.1 * G.2^-1 * G.4^-1, G.4^-1 * G.2^-1 * G.3, G.4 * G.1 * G.2 * G.3 ]

Largest orientation-preserving group for genus 3 has order 168
 ... smooth quotient of triangle group [ 2, 3, 7 ]
Largest group with or without reflections for genus 3 has order 336
 ... smooth quotient of full triangle group [ 2, 3, 7 ]*
Largest group with reflections for genus 3 has order 336
 ... smooth quotient of full triangle group [ 2, 3, 7 ]*
Largest chiral group for genus 3 has order 24
 ... smooth quotient of quadrangle group [ 2, 2, 2, 3 ]
Largest non-orientable group for non-orientable genus 4 has order 48
 ... smooth quotient of full triangle group [ 2, 4, 6 ]*
Check totals true : 47 : 10 chiral, 37 reflexible, plus 8 non-orientable


Genus 4 actions:

Genus 4 action of group of order 72 via triangle group [ 2, 3, 12 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.1 * G.3 * G.2 * G.1 * G.3 * G.2 * G.1 * G.2^-1 * G.3^-1, G.1 * G.3^3 * G.2^2 * G.3^-4 ]

Genus 4 action of group of order 120 via triangle group [ 2, 4, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.3^-2 * G.1 * G.2 * G.3^-2, G.1 * G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1 * G.2^-1 * G.3^-1 ]

Genus 4 action of group of order 72 via triangle group [ 2, 4, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1, G.3^6 ]

Genus 4 action of group of order 40 via triangle group [ 2, 4, 10 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-6 * G.1 * G.2 * G.3^-3 ]

Genus 4 action of group of order 32 via triangle group [ 2, 4, 16 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-4 * G.1 * G.2^-1 * G.3^-3 ]

Genus 4 action of group of order 60 via triangle group [ 2, 5, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^-5, G.3^-2 * G.1 * G.2 * G.3^-2, (G.3^-1 * G.2)^3 ]

Genus 4 action of group of order 36 via triangle group [ 2, 6, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, (G.2 * G.3^-1)^2, G.2^6, G.3^6 ]

Genus 4 action of group of order 36 via triangle group [ 2, 6, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.2^-2 * G.3^-1 * G.2, G.2^6, G.3^6 ]

Genus 4 action of group of order 36 via triangle group [ 2, 6, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.3 * G.2^-1 * G.3^-2, G.2^6, (G.2 * G.3^-1 * G.2)^2 ]

Genus 4 action of group of order 24 via triangle group [ 2, 6, 12 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.2^-2 * G.3^-1 * G.2, G.1 * G.3 * G.2^-1 * G.3^-2, G.2^6 ]

Genus 4 action of group of order 18 via triangle group [ 2, 9, 18 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-9 ]

Genus 4 action of group of order 20 via triangle group [ 2, 10, 10 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-1 * G.3 * G.2^-7 * G.3 ]

Genus 4 action of group of order 16 via triangle group [ 2, 16, 16 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-7 * G.3 ]

Genus 4 action of group of order 36 via triangle group [ 3, 3, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.3^-2 * G.1^-1 * G.3 * G.2^-1, G.3 * G.1^-1 * G.2^-1 * G.1 * G.2^-1 * G.1 ]

Genus 4 action of group of order 36 via triangle group [ 3, 4, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^4, (G.2 * G.3^-1)^2, G.3^4 ]

Genus 4 action of group of order 24 via triangle group [ 3, 4, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^4, G.3 * G.2 * G.1^-1 * G.3 ]

Genus 4 action of group of order 15 via triangle group [ 3, 5, 15 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.2^-5 ]

Genus 4 action of group of order 18 via triangle group [ 3, 6, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.3^-1 * G.1^-1 * G.2^-1, G.2^-6 ]

Genus 4 action of group of order 18 via triangle group [ 3, 6, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2 * G.1^-1 * G.3, G.2^-6 ]

Genus 4 action of group of order 18 via triangle group [ 3, 6, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.1^-1 * G.2^-1 * G.1 * G.3, G.3^-1 * G.2^2 * G.3^-1 ]

Genus 4 action of group of order 20 via triangle group [ 4, 4, 5 ]
Action is chiral
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, (G.1 * G.2^-1)^2, G.2^4, G.3 * G.2 * G.1 * G.3 ]

Genus 4 action of group of order 20 via triangle group [ 4, 4, 5 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3^2 * G.2^-1 * G.1^-1 * G.3^2 ]

Genus 4 action of group of order 20 via triangle group [ 4, 4, 5 ]
Action is chiral
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, (G.1 * G.2^-1)^2, G.2^4, G.3^-1 * G.2 * G.1 * G.3^-1 ]

Genus 4 action of group of order 16 via triangle group [ 4, 4, 8 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3 * G.2 * G.1^-1 * G.3^2 ]

Genus 4 action of group of order 36 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, (G.2 * G.4^-1)^2, G.4^-3 ]

Genus 4 action of group of order 36 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3, G.4^-1 * G.2 * G.1 * G.2 * G.1 * G.2 * G.4 * G.2 ]

Genus 4 action of group of order 36 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, (G.1 * G.3 * G.1 * G.4)^2 ]

Genus 4 action of group of order 24 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^4, (G.3 * G.1)^3, G.4 * G.1 * G.3 * G.1 * G.4^-1 * G.2 ]

Genus 4 action of group of order 24 via quadrangle group [ 2, 2, 2, 4 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.2 * G.4^-1 * G.1 * G.4, G.3 * G.4^-1 * G.2 * G.4 ]

Genus 4 action of group of order 24 via quadrangle group [ 2, 2, 2, 4 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.3 * G.4^-1 * G.1 * G.4, (G.2 * G.1)^3 ]

Genus 4 action of group of order 24 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^4, (G.2 * G.1)^3, G.4^-1 * G.1 * G.2 * G.1 * G.4 * G.3 ]

Genus 4 action of group of order 20 via quadrangle group [ 2, 2, 2, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^-5 ]

Genus 4 action of group of order 20 via quadrangle group [ 2, 2, 2, 5 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-5 ]

Genus 4 action of group of order 20 via quadrangle group [ 2, 2, 2, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4^-5 ]

Genus 4 action of group of order 16 via quadrangle group [ 2, 2, 2, 8 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^-2 * G.1 * G.4^-2 ]

Genus 4 action of group of order 16 via quadrangle group [ 2, 2, 2, 8 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4 * G.3 * G.1 * G.4^2 ]

Genus 4 action of group of order 16 via quadrangle group [ 2, 2, 2, 8 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4 * G.2 * G.1 * G.4^2 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, G.3^-1 * G.1 * G.3^-1 * G.1 * G.3 * G.1 * G.3 * G.1, G.2 * G.1^-1 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3^-1, G.1 * G.3^-1 * G.1 * G.4^-1, G.3^-3 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3^-1, G.1 * G.3^-1 * G.1 * G.3, G.3^-3, G.2 * G.4^-1 * G.1 * G.4 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3^-1, G.1 * G.3^-1 * G.2 * G.4^-1, G.2 * G.3^-1 * G.1 * G.3, G.3^-3 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3^-1, (G.1 * G.3^-1)^2, G.3^-3, G.4^-3 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.4^-1 * G.2^-1, G.2^-3, G.2^-1 * G.1 * G.2 * G.3, G.1 * G.2 * G.1 * G.2^-1 * G.1 * G.3 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.2^-3, G.4^-1 * G.1 * G.2^-1 * G.1, (G.2, G.4), G.3 * G.1^-1 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.2^-1 * G.1 * G.2 * G.1, G.3 * G.1 * G.2^-1 * G.4^-1, G.2^-1 * G.3 * G.2 * G.3, G.3 * G.4^-1 * G.1 * G.4 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.2 * G.1 * G.2^-1 * G.3, G.3 * G.1 * G.2 * G.4, G.4 * G.1 * G.2 * G.3 ]

Genus 4 action of group of order 18 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, (G.2^-1 * G.1)^2, G.3 * G.1 * G.2 * G.4^-1, (G.2^-1 * G.3)^2, G.4^-3 ]

Largest orientation-preserving group for genus 4 has order 120
 ... smooth quotient of triangle group [ 2, 4, 5 ]
Largest group with or without reflections for genus 4 has order 240
 ... smooth quotient of full triangle group [ 2, 4, 5 ]*
Largest group with reflections for genus 4 has order 240
 ... smooth quotient of full triangle group [ 2, 4, 5 ]*
Largest chiral group for genus 4 has order 24
 ... smooth quotient of quadrangle group [ 2, 2, 2, 4 ]
Largest non-orientable group for non-orientable genus 5 has order 120
 ... smooth quotient of full triangle group [ 2, 4, 5 ]*
Check totals true : 47 : 6 chiral, 41 reflexible, plus 9 non-orientable


Genus 5 actions:

Genus 5 action of group of order 192 via triangle group [ 2, 3, 8 ]
Action is reflexible
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, G.3^8, G.3^3 * G.2 * G.3^-1 * G.1 * G.3 * G.2^-1 * G.1 * G.2^-1 * G.3^-2 * G.1 ]

Genus 5 action of group of order 120 via triangle group [ 2, 3, 10 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^-3, G.3^-1 * G.2^-1 * G.1, (G.2 * G.3^-2 * G.1 * G.3)^2 ]

Genus 5 action of group of order 160 via triangle group [ 2, 4, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.3^-2 * G.1 * G.2 * G.3^-2, (G.2 * G.3^-1)^4 ]

Genus 5 action of group of order 96 via triangle group [ 2, 4, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.3^6, G.3 * G.2 * G.3^-1 * G.1 * G.2^-1 * G.3^-2 * G.1 ]

Genus 5 action of group of order 64 via triangle group [ 2, 4, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.1 * G.3 * G.2^-2 * G.3^-2 * G.2, G.3^8 ]

Genus 5 action of group of order 64 via triangle group [ 2, 4, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1, (G.2 * G.3^-1)^4 ]

Genus 5 action of group of order 48 via triangle group [ 2, 4, 12 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-12 ]

Genus 5 action of group of order 40 via triangle group [ 2, 4, 20 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^4, (G.2 * G.3^-1)^2, G.3^-6 * G.1 * G.2^-1 * G.3^-3 ]

Genus 5 action of group of order 80 via triangle group [ 2, 5, 5 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^-5, G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1, G.3^-2 * G.1 * G.2 * G.3^-2 ]

Genus 5 action of group of order 48 via triangle group [ 2, 6, 6 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.2^6, (G.2 * G.3^-1 * G.2)^2, G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1 ]

Genus 5 action of group of order 30 via triangle group [ 2, 6, 15 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.2^-2 * G.3^-1 * G.2, G.2^6, G.3^-2 * G.1 * G.2^-1 * G.3^-2 ]

Genus 5 action of group of order 32 via triangle group [ 2, 8, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, G.1 * G.2^-2 * G.3^-1 * G.2, G.1 * G.3 * G.2^-1 * G.3^-2, G.2^8 ]

Genus 5 action of group of order 32 via triangle group [ 2, 8, 8 ]
Action is reflexible
[ G.1^2, G.3^-1 * G.2^-1 * G.1, (G.2 * G.3^-1)^2, G.3 * G.2 * G.1 * G.2^-1 * G.3^-1 * G.1 ]

Genus 5 action of group of order 22 via triangle group [ 2, 11, 22 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-1 * G.3^2 * G.2^-8 ]

Genus 5 action of group of order 24 via triangle group [ 2, 12, 12 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-1 * G.3^3 * G.2^-7 * G.3 ]

Genus 5 action of group of order 20 via triangle group [ 2, 20, 20 ]
Action is reflexible
[ G.1^2, G.1 * G.2^-1 * G.3^-1, G.3^-1 * G.2^-1 * G.1, G.2^-1 * G.3 * G.2^-8 ]

Genus 5 action of group of order 96 via triangle group [ 3, 3, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.3^4, G.3 * G.2^-1 * G.3^-1 * G.1^-1 * G.3 * G.2 * G.1 ]

Genus 5 action of group of order 60 via triangle group [ 3, 3, 5 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^-3, G.1^-1 * G.3 * G.2^-1 * G.1 * G.3^2, G.3^2 * G.2^-1 * G.1^-1 * G.3^2 ]

Genus 5 action of group of order 48 via triangle group [ 3, 4, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^4, G.3^4, G.2 * G.3^-1 * G.1^-1 * G.2 * G.3^-1 ]

Genus 5 action of group of order 48 via triangle group [ 3, 4, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^4, G.3^4, G.3^-1 * G.2 * G.1^-1 * G.2^2 ]

Genus 5 action of group of order 48 via triangle group [ 3, 4, 4 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.2^4, G.3^4, G.2 * G.3^-1 * G.1^-1 * G.2^-1 * G.3 ]

Genus 5 action of group of order 24 via triangle group [ 3, 6, 6 ]
Action is reflexible
[ G.1^-3, G.2^-1 * G.1^-1 * G.3^-1, G.1^-1 * G.2^-1 * G.1 * G.3, (G.2 * G.3^-1)^2 ]

Genus 5 action of group of order 32 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, G.2^4, G.3^4, G.2 * G.1^-2 * G.3 * G.1^-1, G.3^-1 * G.2 * G.1^-1 * G.2^2 ]

Genus 5 action of group of order 32 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, G.2^4, (G.2 * G.3^-1)^2, G.3^4, G.2 * G.1^-2 * G.3 * G.1^-1 ]

Genus 5 action of group of order 32 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, G.2^4, (G.1 * G.3^-1)^2, G.3^4, G.3^-1 * G.2 * G.1^-1 * G.2^2 ]

Genus 5 action of group of order 32 via triangle group [ 4, 4, 4 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.1^4, (G.1 * G.2^-1)^2, G.2^4, G.3^4, G.3^-2 * G.1^-1 * G.3 * G.2^-1 ]

Genus 5 action of group of order 24 via triangle group [ 4, 4, 6 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3^6 ]

Genus 5 action of group of order 20 via triangle group [ 4, 4, 10 ]
Action is reflexible
[ G.2^-1 * G.1^-1 * G.3^-1, G.3 * G.1^-1 * G.2^-1, G.1^4, G.3^2 * G.2^-1 * G.1 * G.3^2 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, (G.1 * G.3)^4 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^-3, G.1 * G.4 * G.2 * G.1 * G.2 * G.4^-1 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.4^-3, G.1 * G.4 * G.3 * G.1 * G.3 * G.4^-1 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, G.4 * G.3 * G.1 * G.4 * G.2 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-3, G.1 * G.4 * G.1 * G.3 * G.1 * G.4^-1 * G.2 * G.4 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3, (G.1 * G.2)^4 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.4^-3, G.4^-1 * G.1 * G.3 * G.1 * G.4 * G.3 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^-3, G.4^-1 * G.2 * G.4^-1 * G.2 * G.1 * G.2 * G.4 * G.3 ]

Genus 5 action of group of order 48 via quadrangle group [ 2, 2, 2, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^-3, G.4^-1 * G.2 * G.3 * G.2 * G.4 * G.3 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^4, G.1 * G.4 * G.3 * G.1 * G.3 * G.4^-1, (G.1 * G.3)^4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, (G.2 * G.4^-1)^2, G.4^4, G.4^-2 * G.2 * G.1 * G.4 * G.3 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^4, G.4^-1 * G.1 * G.2 * G.1 * G.4 * G.2, (G.1 * G.2)^4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4^-1, G.4^4, (G.1 * G.2)^4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.2 * G.4^-1 * G.2 * G.4, G.4^4, G.3 * G.1 * G.2 * G.1 * G.3 * G.2 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.4^4, (G.1 * G.3)^4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.2 * G.4^-1)^2, G.4^4, G.3 * G.1 * G.2 * G.1 * G.3 * G.2 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.2 * G.4^-1 * G.2 * G.4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.4^4, G.1 * G.3 * G.1 * G.3 * G.1 * G.3 * G.2 * G.4^-1 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.4^4, G.1 * G.4 * G.3 * G.1 * G.3 * G.4^-1, G.1 * G.3 * G.1 * G.3 * G.1 * G.3 * G.2 * G.4 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^4, G.4^-1 * G.1 * G.2 * G.1 * G.4 * G.2, G.1 * G.2 * G.1 * G.2 * G.1 * G.2 * G.4 * G.3 ]

Genus 5 action of group of order 32 via quadrangle group [ 2, 2, 2, 4 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4^-1, G.4^4, G.1 * G.2 * G.1 * G.2 * G.1 * G.2 * G.4 * G.3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4^-1, G.3 * G.4^-1 * G.2 * G.4 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4^-1, G.4 * G.3 * G.2 * G.4, (G.1 * G.2)^4 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.2 * G.4^-1 * G.2 * G.4, G.3 * G.4^-1 * G.1 * G.4, G.3 * G.1 * G.2 * G.1 * G.3 * G.2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is chiral
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4 * G.3 * G.1 * G.4, G.2 * G.4^-1 * G.2 * G.4 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.2 * G.4^-1 * G.1 * G.4 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, G.4 * G.2 * G.1 * G.4, (G.1 * G.3)^4 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^6 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^6 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 2, 6 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4^6 ]

Genus 5 action of group of order 20 via quadrangle group [ 2, 2, 2, 10 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, (G.1 * G.3)^2, G.4^2 * G.3 * G.2 * G.4^2 ]

Genus 5 action of group of order 20 via quadrangle group [ 2, 2, 2, 10 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^2, (G.1 * G.2)^2, G.3 * G.2 * G.1 * G.4^-1, G.1 * G.3 * G.2 * G.4, G.4^2 * G.3 * G.1 * G.4^2 ]

Genus 5 action of group of order 20 via quadrangle group [ 2, 2, 2, 10 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, G.3^2, G.3 * G.2 * G.1 * G.4^-1, G.4^-1 * G.2 * G.1 * G.3, (G.1 * G.3)^2, G.4^2 * G.2 * G.1 * G.4^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, (G.3 * G.1 * G.3^-1 * G.1)^2, G.2 * G.1^-1 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.2 * G.3^-1 * G.1 * G.4^-1, G.3^-3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.2 * G.3^-1 * G.1 * G.3, G.3^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.1 * G.3^-1 * G.1 * G.4^-1, G.3^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.1 * G.3^-1 * G.1 * G.3, G.3^-3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.4^-1 * G.3^-1, G.3^-3, (G.3^-1 * G.1)^4, G.2 * G.1^-1 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible and there is a non-orientable quotient
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, (G.1 * G.3^-1)^2, G.3^-3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, (G.1 * G.2)^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.1 * G.3^-1 * G.2 * G.4, G.3^-3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.1 * G.3^-1 * G.1 * G.4^-1, G.3^-3, (G.2 * G.4^-1)^2, (G.3 * G.4^-1)^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is reflexible
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, G.1 * G.3^-1 * G.2 * G.4, G.3^-3, (G.3 * G.4^-1)^2, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 2, 3, 3 ]
Action is chiral
[ G.1^2, G.2^2, G.3^-1 * G.2 * G.1 * G.4^-1, (G.1 * G.3^-1)^2, G.2 * G.3^-1 * G.2 * G.4^-1, G.3^-3, (G.3 * G.4^-1)^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.3 * G.1 * G.2 * G.4, G.4 * G.1 * G.2 * G.3, G.2^-1 * G.3 * G.2 * G.3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.4^-1 * G.2^-1, G.2^-3, G.2^-1 * G.1 * G.2 * G.3, (G.1 * G.3)^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.2^-3, G.4^-1 * G.1 * G.2^-1 * G.1, (G.2^-1 * G.4^-1)^2, G.3 * G.1^-1 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.2 * G.1 * G.2^-1 * G.3, G.4 * G.1 * G.2 * G.3, (G.1 * G.3)^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is reflexible
[ G.1^2, G.3^2, G.2^-3, G.2^-1 * G.1 * G.2 * G.1, G.3 * G.1 * G.2^-1 * G.4^-1, G.2 * G.3 * G.1 * G.4, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.2^-3, (G.2^-1 * G.1)^2, G.3 * G.1 * G.2 * G.4^-1, (G.2^-1 * G.4^-1)^2, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.4^-1 * G.2^-1, G.2^-3, G.2^-1 * G.1 * G.2 * G.3, (G.2^-1 * G.3 * G.1)^2 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.2^-3, G.4 * G.1 * G.2 * G.3, (G.2^-1 * G.3)^2, (G.2^-1 * G.4^-1)^2, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.2^-3, G.4^-1 * G.1 * G.2^-1 * G.1, (G.2 * G.4^-1)^2, G.3 * G.1^-1 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.2^-3, G.3 * G.1 * G.2^-1 * G.4, G.4 * G.1 * G.2 * G.3, G.4^-3 ]

Genus 5 action of group of order 24 via quadrangle group [ 2, 3, 2, 3 ]
Action is chiral
[ G.1^2, G.3^2, G.2^-3, (G.2^-1 * G.1)^2, G.3 * G.1 * G.2 * G.4^-1, G.2^-1 * G.3 * G.1 * G.4, G.4^-3 ]

Largest orientation-preserving group for genus 5 has order 192
 ... smooth quotient of triangle group [ 2, 3, 8 ]
Largest group with or without reflections for genus 5 has order 384
 ... smooth quotient of full triangle group [ 2, 3, 8 ]*
Largest group with reflections for genus 5 has order 384
 ... smooth quotient of full triangle group [ 2, 3, 8 ]*
Largest chiral group for genus 5 has order 48
 ... smooth quotient of quadrangle group [ 2, 2, 2, 3 ]
Largest non-orientable group for non-orientable genus 6 has order 160
 ... smooth quotient of full triangle group [ 2, 4, 5 ]*
Check totals true : 83 : 20 chiral, 63 reflexible, plus 19 non-orientable