This is a collection of groups that may not fit into some of the other infinite families described elsewhere.
The quaternion group as a set of \(2\times 2\) matrices over \(GF(3)\).
A matrix group consisting of \(2\times 2\) matrices with elements from the finite field of order 3. The group is the quaternion group, the nonabelian group of order 8 that is not isomorphic to the group of symmetries of a square (the dihedral group \(D_4\)).
This group is most easily available via groups.matrix.QuaternionGF3().
The generators are the matrix representations of the elements commonly called \(I\) and \(J\), while \(K\) is the product of \(I\) and \(J\).
sage: from sage.groups.misc_gps.misc_groups import QuaternionMatrixGroupGF3 sage: Q = QuaternionMatrixGroupGF3() sage: Q.order() 8 sage: aye = Q.gens(); aye [1 1] [1 2] sage: jay = Q.gens(); jay [2 1] [1 1] sage: kay = aye*jay; kay [0 2] [1 0]
sage: groups.matrix.QuaternionGF3() Matrix group over Finite Field of size 3 with 2 generators ( [1 1] [2 1] [1 2], [1 1] ) sage: Q = QuaternionMatrixGroupGF3() sage: QP = Q.as_permutation_group() sage: QP.is_isomorphic(QuaternionGroup()) True sage: H = DihedralGroup(4) sage: H.order() 8 sage: QP.is_abelian(), H.is_abelian() (False, False) sage: QP.is_isomorphic(H) False