Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C29 |
| direct product | C4×C61×#577 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C4 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C23×D4 |
| prime radical | C2×C61×#577 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C4×#41×C61 ◁ G | |
2742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 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);(5,6,9,13)(7,11,14,18)(8,10,16,12)(15,17,20,19);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53160);