Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#7199 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C22⋊C4 |
| commutator | C22 | C4×C16×C61 |
| Frattini, cosocle | C23×C8 | C22×C61 |
| socle | C22×C61 | #336 |
| prime radical | C24×C61 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C23×C8×C61 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,11,16,20,15,19,13,18,23,22,24,21)(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)(9,11,13,21,14,19,24,17,15,22,12,20,23,10,16,18)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 187);