Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60316 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C25 |
| commutator | C22 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C24×C4 |
| prime radical | C61×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ G | |
1854 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)(3,7)(4,6)(5,8)(9,10,14,12,13,16,22,21)(11,17,20,24,18,15,23,19)(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);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,17)(21,24)(22,23);(3,8)(5,7);(2,6)(5,7)(11,18)(15,17)(19,24)(20,23);(9,12,14,16,13,21,22,10)(11,19,20,17,18,24,23,15)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53304);