Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#12650 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #442 |
| commutator | C8 | C2×C42×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C2×#135×C61 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×#97×C61 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2)(3,4)(5,6,10,14,12,15,11,18)(7,9,16,20,17,8,13,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(2,4)(5,7,9,14,11,16,19,15,12,17,8,18,10,13,20,6);(5,8,10,19,12,9,11,20)(6,13,14,7,15,16,18,17)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5638);