Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61178 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C61×#525 ◁ G | |
2090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2,4,6)(3,7,8,5)(11,19)(12,13)(15,22)(16,18)(20,21)(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)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24);(9,11,14,20)(10,15,17,23)(12,19,13,21)(16,22,18,24);(3,8)(5,7)(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24);(10,17)(15,23)(16,18)(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, 54166);