Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C27 |
| direct product | C53×#34604 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C53 |
| socle | C23×C53 | C25 |
| prime radical | C24×C53 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(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);(10,13)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(2,5)(6,8)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31);(1,3,4,7)(2,6,5,8)(18,24)(22,30)(23,32)(29,31);(9,11,12,15)(10,14,13,16)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 27592);