Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#8295 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C2×D4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C4×C4⋊C4 |
| prime radical | C23×C53 | C4×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,5,7,4,6,8,3)(9,10,13,18,14,16,12,17)(11,19,22,15,20,23,21,24)(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);(2,6)(3,7)(9,11,12,22,14,20,13,21)(10,15,17,23,16,24,18,19);(1,3,8,6,4,7,5,2)(10,16)(11,20)(17,18)(21,22)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1283);