Is there a library of finite groups given by their multiplication tables? can I get this result using the GAP SYSTEM ?
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$\begingroup$GAP indeed has this functionality in-built. For example, this code will print out Cayley tables corresponding to the two groups of order 6.
n:=6;;
k:=NrSmallGroups(n);;
Print("There are ",k," non-isomorphic groups of order ",n,"\n\n");
for G in AllSmallGroups(6) do Print("Inspecting group: ",StructureDescription(G),"\n"); M:=MultiplicationTable(G); Display(M);
od;It outputs:
There are 2 non-isomorphic groups of order 6
Inspecting group: S3
[ [ 1, 2, 3, 4, 5, 6 ], [ 2, 1, 4, 3, 6, 5 ], [ 3, 6, 5, 2, 1, 4 ], [ 4, 5, 6, 1, 2, 3 ], [ 5, 4, 1, 6, 3, 2 ], [ 6, 3, 2, 5, 4, 1 ] ]
Inspecting group: C6
[ [ 1, 2, 3, 4, 5, 6 ], [ 2, 1, 4, 3, 6, 5 ], [ 3, 4, 5, 6, 1, 2 ], [ 4, 3, 6, 5, 2, 1 ], [ 5, 6, 1, 2, 3, 4 ], [ 6, 5, 2, 1, 4, 3 ] ]The function SmallGroup(n,i) returns a group in the $i$-th isomorphism class of groups of order $n$. E.g. SmallGroup(6,2).
The details of which groups are available can be found here.
$\endgroup$ $\begingroup$The SmallGroups library works in Gap, Sage, Magma, and a bunch of other computer algebra systems. The linked page shows you exactly which groups it contains. Note that it's unlikely that any better package will come along any time soon as the number of isomorphism classes of groups increases dramatically for $p$-groups (especially $2$-groups) above order $2000$.
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