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

Question

Is there a library of finite groups given by their multiplication tables? can I get this result using the GAP SYSTEM ?

Answer 1

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.