Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C53×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C22×C53 | #342 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C22×#41×C53 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 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);(56,57,59,64)(58,66,67,78)(60,65,71,79)(61,63,70,76)(62,73,74,69)(68,80,81,87)(72,77,83,85)(75,84,86,82);(56,58,60,68,61,69,72,82)(57,62,65,75,63,66,77,80)(59,67,71,81,70,73,83,84)(64,78,79,87,76,74,85,86)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5564);