Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#12559 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×#107 |
| prime radical | C24×C53 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,9,14,11,17,23,16)(7,18,21,33,19,32,35,28)(8,15,10,13,24,30,25,31)(12,22,27,34,26,36,29,20)(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)(5,7,10,22,9,21,24,34,11,19,25,36,23,35,8,20)(6,12,15,28,16,29,31,32,17,26,30,33,14,27,13,18);(7,19)(12,26)(18,32)(20,34)(21,35)(22,36)(27,29)(28,33)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5547);