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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.

More miscellaneous short pages on finite groups
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pyright N.S.Wedd 2008