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To do some computations (such as finding copies of GL(2,3) in a larger of groups), I need presentation of GL(2,3). ( Because: Sylow-2 subgroups of GL(2,3) is semidihedral :$ $. It is easy to find copies of semidihedral groups in bigger groups. Therefore, to get a copy of GL(2,3) in larger group, it is sufficient to get element of order 3. If we have presentation of GL(2,3), the relations will provide some path to get elements of order 3, and will give a copy of GL(2,3). ) Can anyone help?

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up vote 3 down vote accepted

You can compute a presentation of any finite group of moderately small order on any generating set using GAP or Magma. But the resulting presentations will not always be illuminating.

${\rm GL}(2,3)$ is of course a 2-generator group, so you might prefer a presentation on two generators.

But a presentation that exhibits its structure as a semidirect product of $Q_8$ and $D_6$ is

Generators: $a,b,c,d$.

Relations: $a^2=b^2$, $b^{-1}ab=a^{-1}$, $c^3=1$, $d^2=1$, $(cd)^2=1$, $c^{-1}ac=b^{-1}$, $c^{-1}bc=a^{-1}b$, $d^{-1}ad=b$.

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