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Cached December 10, 2008, from a web page by Nick Wedd: http://www.weddslist.com/groups/misc/gl23.html. The pictures are taken from a 1985 note I wrote (see Wedd's link below). The remarks are those of Wedd; I do not vouch for their accuracy.

GL(2,3)

GL(2,3) is the group of all 2×2 matrices whose elements are from the three-member ring ℤ₃. 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 C₂²⋊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 C₂²⋊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.

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