Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#22537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C22×D4×C61 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C4×Q8×C61 ◁ G | |
1002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,2,5,6)(3,7,4,8)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(14,20)(17,23);(1,4,5,3)(2,7,6,8)3355 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15525);