Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#29134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×D4×C53 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C53×#376 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,24,21,26,20,25)(19,23,29,32,27,31,28,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,30)(26,31);(3,7)(5,8)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(9,11,12,16)(10,14,13,15)(19,27)(23,31)(28,29)(30,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 22122);