Magma V2.21-7     Wed Mar 23 2016 18:26:44 on Davids-MacBook-Pro-2 [Seed = 
3715666875]

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Type ? for help.  Type <Ctrl>-D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(28,3);
> RunExample(G,6,[2,2,2,7]);
Set seed to 0.


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


--------------------------------------------------
Class |   1  2  3  4    5    6    7    8    9   10
Size  |   1  1  7  7    2    2    2    2    2    2
Order |   1  2  2  2    7    7    7   14   14   14
--------------------------------------------------
p  =  2   1  1  1  1    6    7    5    5    6    7
p  =  7   1  2  3  4    1    1    1    2    2    2
--------------------------------------------------
X.1   +   1  1  1  1    1    1    1    1    1    1
X.2   +   1 -1 -1  1    1    1    1   -1   -1   -1
X.3   +   1  1 -1 -1    1    1    1    1    1    1
X.4   +   1 -1  1 -1    1    1    1   -1   -1   -1
X.5   +   2  2  0  0   Z1 Z1#2 Z1#3 Z1#3   Z1 Z1#2
X.6   +   2  2  0  0 Z1#3   Z1 Z1#2 Z1#2 Z1#3   Z1
X.7   +   2 -2  0  0 Z1#2 Z1#3   Z1  -Z1-Z1#2-Z1#3
X.8   +   2  2  0  0 Z1#2 Z1#3   Z1   Z1 Z1#2 Z1#3
X.9   +   2 -2  0  0   Z1 Z1#2 Z1#3-Z1#3  -Z1-Z1#2
X.10  +   2 -2  0  0 Z1#3   Z1 Z1#2-Z1#2-Z1#3  -Z1


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

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

Z1     = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 0, 1, 0, 
0, 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 7      
        Rep G.1 * G.2

[4]     Order 2       Length 7      
        Rep G.1

[5]     Order 7       Length 2      
        Rep G.3

[6]     Order 7       Length 2      
        Rep G.3^2

[7]     Order 7       Length 2      
        Rep G.3^3

[8]     Order 14      Length 2      
        Rep G.2 * G.3^3

[9]     Order 14      Length 2      
        Rep G.2 * G.3

[10]    Order 14      Length 2      
        Rep G.2 * G.3^2


Surface kernel generators:  [ G.2, G.1 * G.3^2, G.1 * G.2 * G.3, G.3 ]
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, 11)(10, 12)(13, 14)
> g(Inverse(f)(G.1*G.3^2));
(1, 6)(2, 8)(5, 10)(7, 12)(9, 13)(11, 14)