# where can I find a library of finite groups with their multiplication tables?

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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What do you mean "given by their multiplication tables"? Do you just want a list of multiplication tables? I've never used GAP before, but maybe their small groups library along with the PrintTable command is what you're looking for? gap-system.org/Packages/sgl.html groupprops.subwiki.org/wiki/… –  dls Dec 2 '11 at 3:34
PrintTable is not working –  Jawad Dec 2 '11 at 4:50
It's in the Sonata package. gap-system.org/Manuals/pkg/sonata/htm/ref/CHAP001.htm#SECT002 –  dls Dec 2 '11 at 6:18

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.

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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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just curious which examples of "a bunch of other computer algebra systems" are known - the official homepage of the library mentions only GAP and Magma ... –  Alexander Konovalov Apr 23 '13 at 11:09