Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61513 |
| 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×#529 ◁ G | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 85 points:
(2,5)(6,8)(9,10,13,16)(11,18,20,24)(12,17,14,21)(15,22,23,19)(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);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,21)(13,16)(14,17)(18,19)(20,23)(22,24);(9,11,13,20)(10,15,16,23)(12,19,14,22)(17,24,21,18);(2,5)(6,8)(10,16)(15,23)(17,21)(18,24);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,16,21)(11,19,20,22)(15,24,23,18)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54501);