Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C22×C42 |
| direct product | C53×#20470 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,19)(15,16)(22,26)(23,24)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(6,9)(10,12)(13,14,18,22,17,19,25,26)(15,20,23,27,16,21,24,28);(1,3,4,2)(13,15)(14,20)(16,17)(18,23)(19,21)(22,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,21,19,20)(18,24,25,23)(22,28,26,27)5300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13458);