Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C53×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C53 | C23 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C23×C53 | C2×C42 |
| prime radical | C24×C53 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C53 ◁ G | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 87 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);(56,57,59,63,60,62,66,69)(58,61,65,67,64,68,70,71)(72,73)(74,77)(75,79)(76,78)(80,84)(81,83)(82,85)(86,87);(56,58)(57,61)(59,65)(60,64)(62,68)(63,67)(66,70)(69,71)(72,74,75,80,76,81,82,86)(73,77,79,84,78,83,85,87);(57,62)(58,64)(63,69)(65,70)(73,78)(77,83)(79,85)(84,87)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10310);