Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#10842 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C53 | C2×D4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C53 ◁ C22×C42×C53 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4)(10,12)(13,14)(15,23)(16,22)(17,20)(18,21)(19,24)(25,28)(26,27)(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);(5,6,7,8)(13,15,18,26,17,25,16,24)(14,19,22,28,20,27,21,23);(9,10,11,12)(14,20)(19,27)(21,22)(23,28)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3830);