Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×#102 |
| prime radical | C23×C53 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,4,6)(3,7,8,5)(9,10,15,20,16,21,14,31)(11,22,26,32,27,37,25,40)(12,28,29,35,30,19,13,18)(17,23,33,38,34,39,36,24)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(9,11,13,24,15,26,12,23,16,27,29,38,14,25,30,39)(10,17,28,37,31,36,18,32,21,34,19,22,20,33,35,40);(1,3,4,8)(2,5,6,7)(9,12,14,13,16,30,15,29)(10,18,20,28,21,35,31,19)(11,23,25,24,27,39,26,38)(17,32,33,37,34,40,36,22)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6273);