Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(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);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(20,27)(22,28)(25,31)(29,32);(4,5)(8,13)(10,15)(11,14)(17,18)(19,23)(20,28)(21,24)(22,27)(25,32)(26,30)(29,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,21,26)(18,23,24,30)(20,25,27,31)(22,29,28,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12057);