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 matrices over
.
OUTPUT:
A matrix group consisting of 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
).
Note
This group is most easily available via groups.matrix.QuaternionGF3().
EXAMPLES:
The generators are the matrix representations of the
elements commonly called and
, while
is the product of
and
.
sage: from sage.groups.misc_gps.misc_groups import QuaternionMatrixGroupGF3
sage: Q = QuaternionMatrixGroupGF3()
sage: Q.order()
8
sage: aye = Q.gens()[0]; aye
[1 1]
[1 2]
sage: jay = Q.gens()[1]; jay
[2 1]
[1 1]
sage: kay = aye*jay; kay
[0 2]
[1 0]
TESTS:
sage: groups.matrix.QuaternionGF3()
Matrix group over Finite Field of size 3 with 2 generators:
[[[1, 1], [1, 2]], [[2, 1], [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