Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#10940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #386 |
| commutator | C23 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(2,4)(5,6)(7,11,16,23)(8,12)(9,20)(10,13)(14,21)(15,27,28,22)(17,31,30,24)(18,26)(19,33)(25,32)(29,34,36,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(5,7,9,17,10,16,21,30)(6,11,14,24,13,23,20,31)(8,15,19,29,18,28,32,36)(12,22,25,34,26,27,33,35);(1,2,3,4)(5,8,10,18)(6,12,13,26)(7,15,16,28)(9,19,21,32)(11,22,23,27)(14,25,20,33)(17,29,30,36)(24,34,31,35)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3928);