Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C53×#1746 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | C23 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ G | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 83 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);(60,61)(62,65)(63,64)(66,67)(68,69,71,74,72,75,78,81)(70,73,77,80,76,79,82,83);(56,57,58,59)(61,64)(65,67)(68,70)(69,73)(71,77)(72,76)(74,80)(75,79)(78,82)(81,83);(60,62,63,66)(61,65,64,67)(70,76)(73,79)(77,82)(80,83)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13351);