Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#26862 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C4×C4⋊C4×C53 ◁ G | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,15,12,14)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30)(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);(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(18,23)(22,25)(24,32)(30,31)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 19850);