[GAP Forum] The order of Out(S4(8))

[Alexander Hulpke]

Dear Forum, Dear Anvita,

Thank you very much for the bug report. This bug will be fixed in the next release of GAP.

Let me briefly explain what happened, as this indeed looks weird: In GAP 4.7, the algorithm for automorphism groups of simple groups uses a table of automorphism group orders (ATLAS, page xvi) to be able to terminate quickly once all automorphisms were found. This typically produces a very subtantial speedup, as most time is spent in verifying there are no further automorphisms.

Now the group in question is PSp(4,8) which in Lie-language is C2(8). However this group is isomorphic to B2(8) which would be O5(8). As the ATLAS  describes the group as Sp4, GAP also does so, i.e. it recognizes the group as C2(8).

The ATLAS table now gives an exception (the missing 2) for the automorphism group of B2 and does not repeat this exception for C2 explicitly (and only says Cn for n>=3). This special case was not covered, and GAP deduced the wrong automorphism group size (outer 3, not 3*2) and stopped once it was found. This is corrected now.

If you call (after defining S)

gap> SetDataAboutSimpleGroup(S,
> rec( allExtensions := [ [ 2, "2" ], [ 3, "3" ], [ 6, "6" ] ], classicalId := rec( parameter := [ 5, 8 ], series := "O" ), 
>   fullAutGroup := [ 6, "6" ], idSimple := rec( name := "B(2,8) = O(5,8) ~ C(2,8) = S(4,8)", parameter := [ 2, 8 ], 
>       series := "B" ), tomName := "O5(8)" ));

you explicitly set this correct data and the automorphism group calculation will perform fine.

Apologies for the problem!

   Alexander