Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#26861 |
| 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 | |
620 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)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31)(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)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 19849);