Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,23)(19,27)(20,21)(25,26)(28,29)(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);(4,13)(7,12)(9,16)(11,14)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,19,22,29)(18,24,26,32)(20,27,21,28)(23,30,25,31);(17,20,22,21)(18,25,26,23)(19,28,29,27)(24,30,32,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12069);