Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#61038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C4×D4×C61 ◁ G | |
2790 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)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,16)(19,20)(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);(5,7)(6,11)(8,17)(9,19)(10,15)(12,16)(13,20)(14,18);(2,4)(7,15)(11,20)(14,16)(17,19);(5,6)(7,16)(8,12)(9,18)(10,13)(11,19)(14,15)(17,20);(5,8,10,9)(6,12,13,18)(7,17,15,19)(11,16,20,14)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54026);