Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C24 |
| direct product | Q8×M32×C53 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C24 |
| commutator | C22 | C23×C8×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C22×C53 | C23×C8 |
| prime radical | C23×C53 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ G | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,4,6)(3,7,8,5)(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);(1,3,4,8)(2,5,6,7);(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(10,14)(15,20)(16,21)(22,24)5300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26465);