Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C61×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C61 | C23 |
| commutator | C22 | C22×C42×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle, prime radical | C23×C61 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C61 ◁ G | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4)(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);(5,6)(7,12)(8,10)(9,11)(13,14,16,15,17,18,20,19);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,15,16,18,17,19,20,14)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13505);