Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#22315 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C23×C53 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,4,6)(3,7,8,5)(10,12)(11,15)(17,18,20,19,21,23,24,22)(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);(9,10,13,12)(11,16,15,14)(17,19,20,23,21,22,24,18);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,24)(18,19,23,22);(9,12,13,10)(11,16,15,14)(17,20,21,24)(18,22,23,19)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15303);