Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60328 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C25 |
| commutator | C22 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C24×C4 |
| prime radical | C61×#3483 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ G | |
1920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,12,16,21)(11,15,19,22,18,20,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,18)(16,23)(17,22)(21,24);(3,8)(5,7)(9,11)(10,15)(12,20)(13,19)(14,18)(16,23)(17,22)(21,24);(2,6)(5,7)(11,18)(15,20)(19,23)(22,24);(9,12,13,21,14,10,16,17)(11,15,19,22,18,20,23,24)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53316);