//Group G is generated by
//c1c2c4c5(1 2 6 7)(3 4)
//c2c7(1 6)(2 7)
//c1c7(1 7 2 6 5 1)
//(3 4)
//c3c4


10.C2^2:F5
MatrixGroup(11, Integer Ring) of order 2^4 * 5
Generators:

    c1c2c4c5(1 2 6 7)(3 4)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 2  1  0  1  1  0  1  1  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]
    [ 0  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 -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  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]

    c2c7(1 6)(2 7)
    [ 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  0  1  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  0  0  1  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]
    [-1  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]

    c1c7(1 7 2 6 5 1)
    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  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  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  1  0  0  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  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  0  0  0  1]

    (3 4)
    [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 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 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  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:
C2^3
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 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 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 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  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  0  0  0  0  0  0  0  0]
    [ 2  1  0  0  1  1  1  0  1  0  0]
    [ 0  0  1  0  0  0  0  0  0  0  0]
    [ 0  0  0  0  0  0  0  1  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]
    [-1  0  0  0  0  0  0  0 -1  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  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:
F5
MatrixGroup(11, Integer Ring)
Generators:
    [ 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  0  1  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  0  0  1  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]
    [-1  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]

    [ 1  0  0  0  0  0  0  0  0  0  0]
    [ 1  1  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  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  1  0  0  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  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]
    [ 2  1  1  0  0  1  1  1  0  0  0]
    [-1  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  1  0  0  0  0  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  0  0  1  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]