Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C53×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C22⋊C4 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | C4×C22⋊C4 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C23×C8×C53 ◁ G | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
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);(56,57,60,64,61,65,72,78)(58,66,69,80,68,62,73,75)(59,63,70,76,71,77,83,86)(67,79,82,87,81,74,84,85);(56,58)(57,62,65,66)(59,67)(60,69)(61,68)(63,74,77,79)(64,75,78,80)(70,82)(71,81)(72,73)(76,85,86,87)(83,84);(56,59,60,70,61,71,72,83)(57,63,64,76,65,77,78,86)(58,67,69,82,68,81,73,84)(62,74,75,85,66,79,80,87)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13388);