4
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Given the group generated by the matrices

$$\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix},~\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$$

I get a group of order 24, as I have calculated, but which of the 15 possible ones? cf. http://groupprops.subwiki.org/wiki/Groups_of_order_24 It won't be that hard to generate whatever is needed for this group, but since I'm not an expert, what information is sufficient to determine which group it is? This is only one example, I have a bunch of other finite (matrix) groups to determine. TIA.

Here is the multiplication table, generators are 1 and 4, of order 4 and 3 respectively. enter image description here

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    $\begingroup$ How can you distinguish these 15 groups? Can you tell whether this gorup is abelian, what the orders of (some) of its elements are? Etc... $\endgroup$ – Inactive - Objecting Extremism Aug 26 '15 at 18:15
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    $\begingroup$ The first matrix listed is the identity; there's no need to include the identity among a list of generators. $\endgroup$ – Qiaochu Yuan Aug 26 '15 at 18:24
  • $\begingroup$ The order of the generators are 4 and 3 respectively and since they are matrices, the group is not abelian. $\endgroup$ – ArieBos Aug 26 '15 at 19:20
  • $\begingroup$ OK, @Qiaochu Yuan, omitted the identity and added the multiplication table. $\endgroup$ – ArieBos Aug 26 '15 at 19:27
  • $\begingroup$ If you have to do this with several matrix groups, why not try to find its isomorphism class using GAP or something? It's certainly doable by hand (you just need to know enough about all the groups of order $24$ to find what can be used to distinguish them, and look for those characteristics in your group) but it's almost certain to be nontrivial for each group. Some characteristics that might help: center, derived subgroup, conjugacy classes, element orders, that sort of thing. $\endgroup$ – pjs36 Aug 26 '15 at 21:27
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For the example case you could just calculate the derived series to determine which group you are dealing with.

The general case can of course be a lot harder. If your matrix groups are small enough you can e.g. try using GAP and the SmallGroups library.

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  • $\begingroup$ Perfect. Never heard of 'derived' anything, but the derived length appears to be 3 and that with conjugacy class sizes 1,3,6,6,8, makes the group S4. Thanks. $\endgroup$ – ArieBos Aug 27 '15 at 8:29
  • $\begingroup$ I suppose I can ask GAP for the onjugacy calss sizes of the groups with a certain order and a special derived length? $\endgroup$ – ArieBos Aug 27 '15 at 16:49

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