Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#12678 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #442 |
| commutator | C8 | C2×C42×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C2×C61 | #1776 |
| prime radical | C23×C61 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C4×C61 ◁ C2×#97×C61 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,5,12,7,13,21,11,8,14,22,28,6,10,4,20)(3,15,17,26,18,27,31,30,19,29,32,9,23,25,16,24)(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);(1,3,6,18,8,19,7,23)(2,9,10,24,14,26,13,30)(4,17,22,31,21,32,5,16)(11,27,12,29,20,25,28,15);(2,10,14,13)(4,21)(5,22)(9,24,26,30)(11,28,20,12)(15,25,29,27)(16,31)(17,32)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5666);