Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#13297 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#99 |
| commutator | C24 | C22×C4×C61 |
| Frattini, cosocle | C22×D4 | C23×C61 |
| prime radical | C22×D4×C61 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16,12,15,18,23)(11,17,20,24,19,14,21,22)(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);(2,5)(6,7)(9,11,18,20)(10,14,16,22)(12,19,13,21)(15,17,23,24);(1,3,4,8)(2,6,5,7)(10,15)(14,17)(16,23)(22,24)2257 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6285);