Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C2 |
| direct product | C61×#12035 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×D8 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×#107 |
| prime radical | C23×C61 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22)(19,23)(20,26)(24,28)(25,32)(27,31)(29,30)(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)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,23,30,20,25,27,28,21,24,31,32,26,29,19,22);(3,9)(6,12)(10,15)(13,16)(18,24)(22,28)(25,29)(30,32)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5023);