Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#20551 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C24 |
| commutator | C22 | C22×C42×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,21,18,20,17,22)(15,19,24,27,23,26,25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(6,9)(7,11)(13,15,17,25,18,23,16,24)(14,19,22,28,20,26,21,27);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(21,22)(24,25);(6,9)(10,12)(15,23)(19,26)(24,25)(27,28)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13539);