Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#13268 |
| 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 | C23×C61 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C4×C8×C61 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24)(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);(2,5)(6,7)(9,11,12,16,13,19,20,23,14,10,15,18,21,17,22,24);(1,3,4,8)(2,6,5,7)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6256);