Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18963 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3247 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(3,10)(4,5)(7,14)(8,12)(11,13)(15,16)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(19,25)(23,30)(26,31)(29,32);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11951);