Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#29427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×#117×C53 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,24,22,31)(20,29,21,26)(25,30,32,28)(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)(2,5)(4,8)(6,7)(10,13)(14,15)(17,19,23,22)(18,24,27,31)(20,28,21,30)(25,26,32,29);(3,8)(5,7)(17,20,23,21)(18,25,27,32)(19,28,22,30)(24,26,31,29);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,26,27,29)(19,28,22,30)(24,25,31,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 22415);