OFFSET
1,1
COMMENTS
The smallest number n such that there is a simple group and a non-simple perfect group of order n is 20160. So this sequence is A060793 minus A001034 (as sets) for the orders less than 20160. The next known such exceptions are 181440, 262080, 443520 and 604800.
The perfect groups of order 61440, 122880, 172032, 245760, 344064, 491520, 688128, 983040 have not completely been determined yet. Then GAP neither provides the number of these groups nor the groups themselves.
REFERENCES
The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.
LINKS
The GAP Group, GAP Data Library "Perfect Groups".
PROG
(GAP)
OrderPerfectNonSimple:=function(n1, n2)
local it, S, G, L, o, No, i, c;
it:=SimpleGroupsIterator(n1, n2);
S:=[];
for G in it do
Add(S, Order(G));
od;
L:=[];
for o in [n1..n2] do
c:=0;
for i in S do
if i=o then
c:=c+1;
fi;
od;
No:=NumberPerfectGroups(o);
if No>c then
Add(L, o);
if c>0 then
Print([o, c, No]);
fi;
fi;
od;
return L;
end;;
CROSSREFS
KEYWORD
nonn
AUTHOR
Sébastien Palcoux, Sep 29 2019
STATUS
approved