Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#10278 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C22×D4×C53 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2)(3,4)(5,6,10,7)(8,11,9,12)(13,14)(15,23)(16,21)(17,20)(18,22)(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);(2,4)(6,11)(7,12)(8,9)(13,15)(14,19)(16,25)(17,24)(18,26)(20,27)(21,23)(22,28);(5,7,10,6)(8,12,9,11)(14,20)(15,24)(16,17)(21,22)(23,27)(25,26)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3266);