Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13044 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C61 |
| Frattini, cosocle | C22×D4 | C23×C61 |
| prime radical | C2×H(5,𝔽2)×C61 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C23×C61 ◁ C22×D4×C61 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 81 points:
(2,4)(5,6)(7,13,14,19)(8,11,16,18)(9,12)(10,15)(17,20)(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);(1,2,3,4)(7,14)(10,17)(11,18)(15,20);(5,7,9,14)(6,10,12,17)(8,15,16,20)(11,13,18,19)2074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6032);