GL(2,3) is the group of all 2×2 matrices whose elements are from the
three-member ring ℤ_{3}. It is the automorphism group of
C3×C3.

The table below lists all its 48 elements, portraying each as a mapping of the eight non-identity elements of C3×C3, and as a matrix.

The 48 elements of this table form the group **GL(2,3)**, which can be written as
C2↑(C2×C2)⋊C3⋊C2, or as Q8⋊D6.

The 24 pairs of elements enclosed by black lines form a quotient group of order 24,
each such boxed pair being a coset.
This quotient group is PGL(2,3), which can be written as (C2×C2)⋊C3⋊C2,
or as C_{2}^{2}⋊D6.
It is isomorphic with **S4**.

The 24 elements with green backgrounds form a normal subgroup of order 24. This group is
**SL(2,3)**, and can be written as C2↑(C2×C2)⋊C3, or as Q8⋊C3.

The 12 pairs of elements enclosed by black lines *and* having green backgrounds
form a group of order 12, a normal subgroup of PGL(2,3) and a quotient group of SL(2,3).
It can be written as (C2×C2)⋊C3, or as C_{2}^{2}⋊C3.
It is isomorphic with **A4**.

The eight elements with the brighter green backgrounds form **Q8**.

For more information on GL(2,3) see Visualizing GL(2,p) by Steven H. Cullinane.

More miscellaneous short pages on finite groups

More pages on groups

Copyright N.S.Wedd 2008