Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#7936 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| socle | C22×C61 | C2×#95 |
| prime radical | C24×C61 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C42×C61 ◁ G | |
550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18,21,24,22,25,20,29)(19,26,27,31,28,23,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,26)(29,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 924);