Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C29 |
| direct product | D4×#135×C61 |
| 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 | |
2822 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);(2,4);(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)4270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54036);