//Group G is generated by
//c2c3c6c7(1 2)(4 5 6 7)
//c1c2c5c7(4 6)(5 7)
//c5c7(3 4 7 5 6)


3.F5
MatrixGroup(11, Integer Ring) of order 2^2 * 5
Generators:

    c2c3c6c7(1 2)(4 5 6 7)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  1  0  1  1  0  0  1  0  0]
    [ 0  0  0  1  0  0  0  0  0  0  0]
    [-1  0 -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  0  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [-1  0  0  0  0  0  0  0 -1  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  0  0  0  0  0  0  1]

    c1c2c5c7(4 6)(5 7)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  1  1  0  0  1  0  1  0  0]
    [-1  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  1  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  0  0  0]
    [-1  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  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]
    
    c5c7(3 4 7 5 6)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  0  0  0  0  1  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  0  1  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  1  0  0]
    [-1  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 -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]
1
{@
    ( 0  1 -1  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  1  0 -1  0  0  0  0  0  0  0)
@}
Intersection Matrix:
[-1  0  1  0]
[ 0 -1  0  1]
[ 1  0 -1  0]
[ 0  1  0 -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]
    [ 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  1  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  0  1  0  0]
    [-1  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 -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]
2
{@
    ( 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 -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 0 0 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)
@}
Intersection Matrix:
[-1  0  0  0  0  0  0  0  0  1]
[ 0 -1  0  0  0  1  0  0  0  0]
[ 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  1  0  0  0 -1  0  0  0  0]
[ 0  0  0  0  1  0 -1  0  0  0]
[ 0  0  0  0  0  0  0 -1  1  0]
[ 0  0  0  0  0  0  0  1 -1  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]
    [ 1  1  1  1  0  0  0  0  0  0  0]
    [-1  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  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  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]