Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#10868 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #117 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | #382 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#117×C53 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6)(7,16,19,12)(8,21)(9,25)(10,15)(11,14)(13,22)(17,31,33,35)(18,27,20,30)(23,28)(24,36)(26,32,29,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,3,4)(5,7,10,20,11,19,9,18)(6,12,15,27,14,16,25,30)(8,17,23,34,24,33,22,32)(13,26,21,35,28,29,36,31);(5,8,9,22,11,24,10,23)(6,13,15,21,14,28,25,36)(7,17,18,32,19,33,20,34)(12,26,27,35,16,29,30,31)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3856);