Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C26 |
| direct product | C4×C53×#396 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | C23 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| socle | C24×C53 | C24 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ G | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55,56,57);(58,59,60,61)(67,69);(62,63)(64,65)(66,67,68,69);(59,61)(63,65)(66,67)(68,69)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6853);