I am interested in the subgroup structure of the affine general linear group $AGL(2,3)$, in particular I want to know if they could have dihedral subgroups other then $D_3$ and $D_4$, i.e. the ones of order $6$ and $8$ (and $C_2 \times C_2$ if you consider it as a dihedral group). This group could be described as the group of matrices over $\mathbb F_3$ (the finite field containing three elements) of the form $$ A = \begin{pmatrix} e & 0 & 0 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{pmatrix} $$ with $e, a_1, a_2, a_3, b_1, b_2, b_3 \in \mathbb F_3$ and $\operatorname{det}(A) = 1$. This group also appears as the automorphism group of the group of order $27$ and exponent $3$, see this post where it is described slightly different.

So has this group any subgroups which are dihedral groups other then $D_3$ and $D_4$?

  • $\begingroup$ According to GAP, you can also find the dihedral group of order $12$. $\endgroup$ – Mikko Korhonen Jan 30 '16 at 20:14
  • $\begingroup$ @MikkoKorhonen Thanks for that. Would you mind posting this with your gap code as an answer; I am no expert in gap and like to view your example!? PS: I add the tag "gap" to make such an answer more appropriate. $\endgroup$ – StefanH Jan 30 '16 at 20:18
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    $\begingroup$ G := AutomorphismGroup(SylowSubgroup(GL(3,3),3));; L := List(ConjugacyClassesSubgroups(G), StructureDesription(Representative))); $\endgroup$ – Mikko Korhonen Jan 30 '16 at 20:50
  • $\begingroup$ ${\rm GL}(2,3)$ has subgroups $D_2$, $D_3$ and $D_4$. $\endgroup$ – Derek Holt Jan 30 '16 at 20:53
  • $\begingroup$ @DerekHolt What you wanna say with your comment? That every dihedral subgroup of $AGL(2,3)$ gives rise to a dihedral subgroup of $GL(2,3)$? Then $D_6$ may not be possible, as the only groups that could be obvisouly built from subgroups $U \le GL(2,3)$ are $U$ times $\mathbb F_3^2$? $\endgroup$ – StefanH Jan 30 '16 at 21:24

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