Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#13091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#99 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C22×D4 | C23×C61 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C22×#41×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:
(1,2)(3,4)(5,6)(7,14,15,20)(8,11)(9,12,18,19)(10,13)(16,17)(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);(5,7,9,17)(6,8,12,14)(10,15,18,11)(13,16,19,20);(2,4)(5,8,10,16)(6,11,13,17)(7,12,15,19)(9,14,18,20)2257 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6079);