//Group G is generated by
//c1c2c4c5(1 3 7 2)(5 6)
//c1c5c6c7(1 7)(2 3)
//c2c3(2 5 3)
//c6c7(1 7 6)


5.C3:S3.C2
MatrixGroup(11, Integer Ring) of order 2^2 * 3^2
Generators:

    c1c2c4c5(1 3 7 2)(5 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  1  0  1  1  0  1  0  0  0]
    [-1  0  0  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  0  0  1  0  0]
    [-1  0  0  0  0 -1  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]
    [ 0  0  0  1  0  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]

    c1c5c6c7(1 7)(2 3)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  1  0  0  0  1  1  1  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  0  0  0]
    [ 0  0  0  0  0  1  0  0  0  0  0]
    [-1  0  0  0  0  0 -1  0  0  0  0]
    [-1  0  0  0  0  0  0 -1  0  0  0]
    [-1  0 -1  0  0  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]

    c2c3(2 5 3)
    [ 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]
    [-1  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  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]
    
    c6c7(1 7 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  1  0  0  0  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  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]
    [-1  0 -1  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  1]
1
{@
    ( 0  1  0  0  0 -1  0  0  0  0  0),
    (0 0 0 0 0 1 0 0 0 0 0)
@}
Intersection Matrix:
[-1  1]
[ 1 -1]
Stabilizer Group Name:
C3:S3
MatrixGroup(11, Integer Ring)
Generators:
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  1  0  0  0  1  1  1  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  0  0  0]
    [ 0  0  0  0  0  1  0  0  0  0  0]
    [-1  0  0  0  0  0 -1  0  0  0  0]
    [-1  0  0  0  0  0  0 -1  0  0  0]
    [-1  0 -1  0  0  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]

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

    [ 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]
    [-1  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  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]
2
{@
    (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  0  0 -1  0  0  0),
    ( 0  1  0 -1  0  0  0  0  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  0 -1  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  1  0  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  1  0  0  0 -1  0  0  0  0  0  0]
[ 0  0  0  0  0  0 -1  0  1  0  0  0]
[ 0  0  0  0  0  0  0 -1  0  1  0  0]
[ 0  0  0  0  0  0  1  0 -1  0  0  0]
[ 0  0  0  0  0  0  0  1  0 -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:
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]
    [-1  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  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]