Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#61624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C53×#525 ◁ G | |
2156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,14,23)(12,21,13,18)(17,22,24,20)(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)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,23)(12,20,13,22)(17,18,24,21);(9,12,15,13)(10,17,19,24)(11,20,14,22)(16,18,23,21);(3,8)(5,7);(2,6)(5,7)(9,13,15,12)(10,18,19,21)(11,20,14,22)(16,17,23,24)2597 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 54612);