User:Tomruen/424 subgroups

Matrices
The group is order 64, which represent 64 matrices possible from products of the 4 generators, with matrix characteristic structure (1:0/1 2:16/19 4:16/12). It has 1 identity matrix, 16 reflection matrices [], 16 2-fold rotation matrices [2]+, 2 2D central inversions [2+,2+], and one 3D central inversion [2+,2+,2+], 16 order-4 rotoreflective matrices [2+,4+], and 12 rotational 4-fold matrices: 8 [4]+ and 4 double rotations [4+,2+,4+]+.

Subgroups
Generators are listed as a set of matrices, index 0 to 3, or letters for extended groups.

The structure is expressed by looking at all possible matrix products of generators. They are counted by order:A/B, where order of matrix M is how many self-products produce the identity. The A count have determinant -1 (reflective), and B count have determinant +1 (pure rotations). Every group has (1:0/1} for the identity matrix.