Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#11418 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C23 | C22×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C2×C22⋊C8 |
| prime radical | C25×C61 | C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C8×C61 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24)(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);(1,3)(2,5)(4,8)(6,7);(10,14)(15,20)(16,21)(22,24)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4406);