Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C2 |
| direct product | C61×#12025 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×D8 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×#107 |
| prime radical | C23×C61 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,3,6,4,5,7,8)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22)(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);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(2,5)(6,8)(10,16)(14,20)(17,21)(22,24)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5013);