Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#21053 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C24 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C25×C61 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ G | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,4)(6,7)(9,10,13,17,14,11,15,19)(12,16,21,23,20,18,22,24)(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);(5,6,8,7)(9,11,13,19,14,10,15,17)(12,18,21,24,20,16,22,23);(2,4)(5,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(10,11)(12,20)(17,19)(21,22)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 14041);