Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13240 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,4,3)(5,6,7,8)(9,10)(11,22)(12,29)(13,31)(14,32)(15,21)(16,25)(17,26)(18,24)(19,23)(20,30)(27,40)(28,33)(34,37)(35,38)(36,39)(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);(1,3,4,2)(6,8)(9,11,23,37,14,27,12,25,15,28,24,38,30,39,13,26)(10,16,31,40,20,34,18,22,21,17,29,36,32,35,19,33);(9,11,24,38,14,27,13,26,15,28,23,37,30,39,12,25)(10,17,29,40,20,35,19,22,21,16,31,36,32,34,18,33)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6228);