//Group G is generated by
//c1c3(2 5)(4 6)
//c3c4(1 6 2)(3 5 4)
//c2c4(1 3)(2 4)
//c2c4c5c6(2 4)(5 6)


7.S4
MatrixGroup(11, Integer Ring) of order 2^3 * 3
Generators:

    c1c3(2 5)(4 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  1  0  1  0  0  0  0  0  0]
    [-1  0 -1  0  0  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  1  0  0  0  0]
    [-1  0  0  0 -1  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [ 0  0  0  1  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  1  0  0]
    [ 0  0  0  0  0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]

    c3c4(1 6 2)(3 5 4)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  0  1  0  1  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  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]
    [-1  0  0  0 -1  0  0  0  0  0  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  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  1]

    c2c4(1 3)(2 4)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  1  0  1  0  0  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  1  0  0  0  0  0  0  0  0]
    [-1  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  1  0  0  0]
    [ 0  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  1]
    
    c2c4c5c6(2 4)(5 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  0  1  0  1  1  1  0  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  0  0  1  0  0  0  0  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [-1  0  0  0  0  0  0 -1  0  0  0]
    [-1  0  0  0  0  0 -1  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  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]
1
{@
    (0 0 1 0 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),
    ( 0  1  0  0 -1  0  0  0  0  0  0),
    (0 0 0 0 1 0 0 0 0 0 0),
    ( 0  1  0  0  0 -1  0  0  0  0  0),
    ( 0  1  0  0  0  0  0 -1  0  0  0),
    (0 0 0 0 0 1 0 0 0 0 0),
    ( 0  1  0 -1  0  0  0  0  0  0  0),
    (0 0 0 1 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)
@}
Intersection Matrix:
[-1  0  0  0  0  0  0  0  0  0  1  0]
[ 0 -1  0  0  0  0  1  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  0  0  1]
[ 0  0  0 -1  1  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  0  1  0  0  0  0]
[ 0  1  0  0  0  0 -1  0  0  0  0  0]
[ 0  0  0  0  0  1  0 -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  1 -1  0  0]
[ 1  0  0  0  0  0  0  0  0  0 -1  0]
[ 0  0  1  0  0  0  0  0  0  0  0 -1]
Stabilizer Group Name:
C2
MatrixGroup(11, Integer Ring)
Generators:
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  0  1  0  1  1  1  0  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  0  0  1  0  0  0  0  0  0]
    [-1  0  0 -1  0  0  0  0  0  0  0]
    [-1  0  0  0  0  0  0 -1  0  0  0]
    [-1  0  0  0  0  0 -1  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  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]