Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18710 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3251 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(18,22)(20,27)(23,29)(24,28)(25,31)(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);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(20,27)(22,28)(25,31)(29,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,32)(26,30)(29,31);(3,9)(7,14)(10,15)(13,16)(17,19,21,26)(18,23,24,30)(20,25,27,31)(22,29,28,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11698);