Magma V2.20-3     Sun Mar 20 2016 11:34:27 on Fordhamwinarski [Seed = 
1976426535]
Type ? for help.  Type <Ctrl>-D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(20,4);
> RunExample(G,5,[2,2,2,10]);
Set seed to 0.


Character Table of Group G
--------------------------


----------------------------------------
Class |   1  2  3  4    5    6    7    8
Size  |   1  1  5  5    2    2    2    2
Order |   1  2  2  2    5    5   10   10
----------------------------------------
p  =  2   1  1  1  1    6    5    5    6
p  =  5   1  2  3  4    1    1    2    2
----------------------------------------
X.1   +   1  1  1  1    1    1    1    1
X.2   +   1  1 -1 -1    1    1    1    1
X.3   +   1 -1 -1  1    1    1   -1   -1
X.4   +   1 -1  1 -1    1    1   -1   -1
X.5   +   2 -2  0  0   Z1 Z1#2-Z1#2  -Z1
X.6   +   2  2  0  0   Z1 Z1#2 Z1#2   Z1
X.7   +   2  2  0  0 Z1#2   Z1   Z1 Z1#2
X.8   +   2 -2  0  0 Z1#2   Z1  -Z1-Z1#2


Explanation of Character Value Symbols
--------------------------------------

# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k

Z1     = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | -1, 0, -1, 
-1 ]


Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep Id(G)

[2]     Order 2       Length 1      
        Rep G.2

[3]     Order 2       Length 5      
        Rep G.1 * G.2

[4]     Order 2       Length 5      
        Rep G.1

[5]     Order 5       Length 2      
        Rep G.3^2

[6]     Order 5       Length 2      
        Rep G.3

[7]     Order 10      Length 2      
        Rep G.2 * G.3

[8]     Order 10      Length 2      
        Rep G.2 * G.3^2


Surface kernel generators:  [ G.2, G.1 * G.2 * G.3^2, G.1 * G.2 * G.3^3, G.2 * 
G.3^4 ]
Is hyperelliptic?  true
Curve is hyperelliptic
> FP,f:=FPGroup(G);
> PermG,g:=PermutationGroup(FP);
> g(Inverse(f)(G.2));
(1, 2)(3, 4)(5, 7)(6, 8)(9, 10)
> g(Inverse(f)(G.1 * G.2 * G.3^2));
(1, 8)(2, 6)(3, 4)(5, 10)(7, 9)
> g(Inverse(f)(G.1 * G.2 * G.3^3));
(1, 10)(2, 9)(3, 8)(4, 6)(5, 7)