# Miscellaneous Groups¶

This is a collection of groups that may not fit into some of the other infinite families described elsewhere.

sage.groups.misc_gps.misc_groups.QuaternionMatrixGroupGF3()

The quaternion group as a set of $$2\times 2$$ matrices over $$GF(3)$$.

OUTPUT:

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 $$D_4$$).

Note

This group is most easily available via groups.matrix.QuaternionGF3().

EXAMPLES:

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()[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]  [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

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