Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18976 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3554 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,9)(3,11,12,15)(4,7,5,10)(8,13,16,14)(18,23)(19,21)(24,30)(25,29)(26,27)(31,32)(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);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,3,6,12)(2,8,9,16)(4,14,5,13)(7,15,10,11)(17,19,22,21)(18,23,25,29)(20,27,28,26)(24,31,32,30);(2,9)(3,12)(7,10)(13,14)(17,20,22,28)(18,24,25,32)(19,26,21,27)(23,30,29,31)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11964);