Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#32781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C2×#121×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×#117×C53 ◁ G | |
694 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)(9,10,13,15)(11,16,12,14)(17,18)(19,26)(20,29)(21,25)(22,24)(23,28)(27,32)(30,31)(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)(9,11,13,12)(10,14,15,16)(19,27)(20,22)(23,31)(24,29)(26,32)(28,30);(2,6)(5,8)(10,15)(11,12)(17,19)(18,23)(20,27)(21,28)(22,30)(24,32)(25,26)(29,31);(3,7)(5,8)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,32,26,31)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 25769);