Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#32733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×#117×C53 ◁ G | |
682 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,21,25)(19,26,29,22)(20,23,31,24)(27,28,32,30)(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)(2,5)(4,7)(6,8)(10,13)(11,16)(17,19,21,29)(18,22,25,26)(20,30,31,28)(23,32,24,27);(2,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,31)(18,23,25,24)(19,27,29,32)(22,30,26,28);(3,7)(5,8)(11,16)(14,15)(17,20)(18,24)(19,28)(21,31)(22,32)(23,25)(26,27)(29,30)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25721);