Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#11411 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C61 | C23 |
| commutator | C22 | C2×C4×C8×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×C4×C8 |
| prime radical | C23×C61 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C61 ◁ G | |
290 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)(23,28)(24,29)(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)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(3,9)(6,12)(10,15)(13,16)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4399);