Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60721 |
| 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 ◁ C4×#41×C61 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,23)(14,17)(15,21)(19,20)(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);(2,5)(6,8)(9,10)(11,19)(12,23)(13,16)(14,17)(15,22)(18,20)(21,24);(9,11,13,21,14,22,12,20)(10,15,16,18,17,19,23,24);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,23)(13,16)(14,17)(18,21)(19,22)(20,24);(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53709);