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×#13286 |
| 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 | |
342 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)(9,10,15,20,16,21,14,32)(11,22,26,36,27,39,25,33)(12,28,30,37,31,19,13,18)(17,29,34,40,35,23,38,24)(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);(9,11,13,24,15,26,12,29,16,27,30,40,14,25,31,23)(10,17,28,39,32,38,18,36,21,35,19,22,20,34,37,33);(1,3,4,8)(2,5,6,7)(9,12,14,13,16,31,15,30)(10,18,20,28,21,37,32,19)(11,23,25,40,27,29,26,24)(17,33,34,22,35,36,38,39)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6274);