//Group G is generated by
//(1 3 6)(4 5 7)
//c1c2c3c4c5c6c7c8(1 5)(3 7)(4 6)
//c2c3(1 5 6 3 2 7 4)


1.F7
MatrixGroup(11, Integer Ring)
Generators:

    (1 3 6)(4 5 7)
    [1 0 0 0 0 0 0 0 0 0 0]
    [0 1 0 0 0 0 0 0 0 0 0]
    [0 0 0 0 1 0 0 0 0 0 0]
    [0 0 0 1 0 0 0 0 0 0 0]
    [0 0 0 0 0 0 0 1 0 0 0]
    [0 0 0 0 0 0 1 0 0 0 0]
    [0 0 0 0 0 0 0 0 1 0 0]
    [0 0 1 0 0 0 0 0 0 0 0]
    [0 0 0 0 0 1 0 0 0 0 0]
    [0 0 0 0 0 0 0 0 0 1 0]
    [0 0 0 0 0 0 0 0 0 0 1]

    c1c2c3c4c5c6c7c8(1 5)(3 7)(4 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 4  1  1  1  1  1  1  1  1  1  0]
    [-1  0  0  0  0  0 -1  0  0  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [-1  0  0  0  0  0  0  0 -1  0  0]
    [-1  0  0  0  0  0  0 -1  0  0  0]
    [-1  0 -1  0  0  0  0  0  0  0  0]
    [-1  0  0  0  0 -1  0  0  0  0  0]
    [-1  0  0  0 -1  0  0  0  0  0  0]
    [-1  0  0  0  0  0  0  0  0 -1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]

    c2c3(1 5 6 3 2 7 4)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  1  0  0  0  0  1  0  0]
    [ 0  0  0  0  0  0  1  0  0  0  0]
    [-1  0  0  0  0  0  0  0 -1  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [ 0  0  1  0  0  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [ 0  0  0  0  1  0  0  0  0  0  0]
    [ 0  0  0  0  0  1  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]
    1
{@
    Mod: ( 0  1 -1  0  0  0  0  0  0  0  0),
    Mod: (0 0 0 0 0 0 0 0 1 0 0),
    Mod: (0 0 0 0 0 0 1 0 0 0 0),
    Mod: ( 0  1  0  0 -1  0  0  0  0  0  0),
    Mod: (0 0 0 0 1 0 0 0 0 0 0),
    Mod: (0 0 0 1 0 0 0 0 0 0 0),
    Mod: ( 0  1  0  0  0  0  0  0 -1  0  0),
    Mod: (0 0 0 0 0 0 0 1 0 0 0),
    Mod: (0 0 1 0 0 0 0 0 0 0 0),
    Mod: ( 0  1  0  0  0  0 -1  0  0  0  0),
    Mod: ( 0  1  0 -1  0  0  0  0  0  0  0),
    Mod: ( 0  1  0  0  0 -1  0  0  0  0  0),
    Mod: (0 0 0 0 0 1 0 0 0 0 0),
    Mod: ( 0  1  0  0  0  0  0 -1  0  0  0)
@}
Intersection Matrix:
[-1  0  0  0  0  0  0  0  1  0  0  0  0  0]
[ 0 -1  0  0  0  0  1  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  1  0  0  0  0]
[ 0  0  0 -1  1  0  0  0  0  0  0  0  0  0]
[ 0  0  0  1 -1  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0 -1  0  0  0  0  1  0  0  0]
[ 0  1  0  0  0  0 -1  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0 -1  0  0  0  0  0  1]
[ 1  0  0  0  0  0  0  0 -1  0  0  0  0  0]
[ 0  0  1  0  0  0  0  0  0 -1  0  0  0  0]
[ 0  0  0  0  0  1  0  0  0  0 -1  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0  0 -1  1  0]
[ 0  0  0  0  0  0  0  0  0  0  0  1 -1  0]
[ 0  0  0  0  0  0  0  1  0  0  0  0  0 -1]
Stabilizer Group Name:
C3
MatrixGroup(11, Integer Ring)
Generators:
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  1  0  0  1  0  0  0  0]
    [ 0  0  1  0  0  0  0  0  0  0  0]
    [-1  0  0  0  0  0 -1  0  0  0  0]
    [ 0  0  0  0  0  1  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  1  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [ 0  0  0  0  1  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]
2
{@
    Mod: (0 0 0 0 0 0 0 0 0 1 0),
    Mod: ( 0  1  0  0  0  0  0  0  0 -1  0)
@}
Intersection Matrix:
[-1  1]
[ 1 -1]
Stabilizer Group Name:
C7:C3
MatrixGroup(11, Integer Ring)
Generators:
    [1 0 0 0 0 0 0 0 0 0 0]
    [0 1 0 0 0 0 0 0 0 0 0]
    [0 0 0 0 1 0 0 0 0 0 0]
    [0 0 0 1 0 0 0 0 0 0 0]
    [0 0 0 0 0 0 0 1 0 0 0]
    [0 0 0 0 0 0 1 0 0 0 0]
    [0 0 0 0 0 0 0 0 1 0 0]
    [0 0 1 0 0 0 0 0 0 0 0]
    [0 0 0 0 0 1 0 0 0 0 0]
    [0 0 0 0 0 0 0 0 0 1 0]
    [0 0 0 0 0 0 0 0 0 0 1]

    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  1  0  0  0  0  1  0  0]
    [ 0  0  0  0  0  0  1  0  0  0  0]
    [-1  0  0  0  0  0  0  0 -1  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [ 0  0  1  0  0  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [ 0  0  0  0  1  0  0  0  0  0  0]
    [ 0  0  0  0  0  1  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]