//Group G is generated by
//c2c4(1 5)(3 4)(6 7)
//c1c2(1 5 7 2 6)
//c3c4


1.C5:C4
MatrixGroup(11, Integer Ring) of order 2^2 * 5
Generators:

    c2c4(1 5)(3 4)(6 7)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  1  1  0  0  0  0  0  0]
    [ 0  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]
    [-1  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  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  0  0  0  0  0  0  1]

    c1c2(1 5 7 2 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  0  0  0  1  1  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]
    [ 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  1  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  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]
    
    c3c4
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  0  1  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]
    [-1  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  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]
1
{@
    (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 1 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  1  0]
[ 0 -1  0  0  0  0  0  0  0  1]
[ 0  0 -1  1  0  0  0  0  0  0]
[ 0  0  1 -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  1  0 -1  0  0  0]
[ 0  0  0  0  0  1  0 -1  0  0]
[ 1  0  0  0  0  0  0  0 -1  0]
[ 0  1  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]
    [ 1  1  0  0  1  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]
    [-1  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  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]
2
{@
    (0 0 0 0 1 0 0 0 0 0 0),
    ( 0  1  0  0 -1  0  0  0  0  0  0),
    ( 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  0  0]
[ 1 -1  0  0]
[ 0  0 -1  1]
[ 0  0  1 -1]
Stabilizer Group Name:
C5
MatrixGroup(11, Integer Ring)
Generators:
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  0  0  0  1  1  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]
    [ 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  1  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  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]