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I hope this is an easy question but no so easy to be wiped out. I have asked few physical people (Profs) here in our department, and nobody has a clue. I'm reading this paper:

The locus of curves with prescribed automorphism group

and it uses following notation to represent a group:

G = (16, 13)

This is at the beginning of page 11. Shaska uses this notation in his other papers but I couldn't find an explanation for the kind of groups he's talking about. Could you please tell me exactly what's the meaning of this bracket notation? It could be a GAP thing but unfortunately I'm not a GAP person.

Thanks a lot.

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    $\begingroup$ This seems to refer to the number of the finite group in the GAP database. The first number is the order of the group and the second is the GAP-specific number of that particular group among all groups of that order. E.g. in MAGMA, this group is created with the command SmallGroup(16,13). $\endgroup$
    – Alex B.
    Commented Feb 22, 2011 at 6:38
  • $\begingroup$ I see. So it's the 13th group of order 16. Is GAP using a particular way to order all groups of order n? $\endgroup$
    – André Henriques
    Commented Feb 22, 2011 at 10:23
  • $\begingroup$ Not a GAP person, yet. $\endgroup$
    – Kevin O'Bryant
    Commented Feb 22, 2011 at 15:50

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If Alex Bartel is correct that the notation $(16,13)$ refers to SmallGroup(16,13), then here are generators for your group: $$ \left\{\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right),\left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right),\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)\right\}$$

It is extremely convenient to calculate such representations in GAP. Here is the code that generated this representation:

IrreducibleRepresentations(SmallGroup(16,13))[10];

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  • $\begingroup$ Thank you very much. I'm going to check GAP/MAGMA to see if I get the same thing. $\endgroup$
    – Syed
    Commented Feb 22, 2011 at 15:56
  • $\begingroup$ I think I got my answer but it rises the new question that what's the logic behind the ordering in small group database. I think I should be able to find my answer if I dig into GAP documents. $\endgroup$
    – Syed
    Commented Feb 22, 2011 at 19:59
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    $\begingroup$ I don't think the ordering is completely canonical, but the outer-most criterion for ordering (for p-groups) is the size of the minimum generating set. So for a p-group, the cyclic group always comes first, then all the subgroups generated by 2 elements, then all the subgroups generated by 3 elements, and so on, and the elementary abelian group comes last. $\endgroup$
    – Vipul Naik
    Commented Feb 26, 2011 at 20:07
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    $\begingroup$ Also, you can get more information on groups of order 16 here: groupprops.subwiki.org/wiki/Groups_of_order_16 $\endgroup$
    – Vipul Naik
    Commented Feb 26, 2011 at 20:07

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