Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#20508 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator | C4 | C22×C42×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C22×D8×C61 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C61×#376 ◁ G | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,3)(9,10,13,16)(11,18,19,15)(12,14,21,17)(20,23,24,22)(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)(10,14)(12,21)(15,22)(16,17)(18,23)(20,24);(5,7,8,6)(9,11)(10,15)(12,20)(13,19)(14,22)(16,18)(17,23)(21,24);(1,3,4,2)(10,16)(11,19)(14,17)(20,24)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13496);