Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#12638 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 77 points:
(5,6,8,7)(9,10)(11,22)(12,17)(13,19)(14,18)(15,21)(16,20)(23,24)(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);(1,2,3,4)(9,11,14,21,12,23,24,17,15,18,22,10,16,19,13,20);(5,7,8,6)(9,12,15,16)(10,17,21,20)(11,19,18,23)(13,14,24,22)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5626);