Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3567 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
838 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)(9,15)(14,16)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,28,26)(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);(4,11)(6,12)(9,15)(13,16)(17,19,22,27)(18,23,20,28)(21,30,31,26)(24,32,29,25);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(21,31)(24,29)(25,32)(26,30);(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,13,12,16)(17,20,22,18)(19,23,27,28)(21,29,31,24)(25,30,32,26)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12029);