Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#102 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,21,25,35)(12,18)(13,29)(14,31)(15,20)(16,23,34,30)(17,27)(19,26)(22,37,39,33)(24,40,38,32)(28,36)(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);(9,11,12,22,26,38,13,30,15,25,28,39,14,24,27,23)(10,16,29,40,31,37,18,35,20,34,17,32,19,33,36,21);(1,3,4,8)(2,5,6,7)(9,12,26,13,15,28,14,27)(10,17,31,36,20,29,19,18)(11,22,38,30,25,39,24,23)(16,32,37,21,34,40,33,35)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6276);