Finite
Geometry Notes |

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.

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.

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Copyright N.S.Wedd 2008