Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C53 · Schur multiplier (H2) = C26 |
| direct product | C2×C53×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C22⋊C4 |
| commutator | C22 | C23×C8×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C22×C53 | C2×C22⋊C8 |
| prime radical | C22×#41×C53 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C23×C8×C53 ◁ G | |
1154 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);(56,57,59,64,60,65,71,79,61,63,72,76,73,77,84,86)(58,66,68,78,69,80,82,87,67,62,70,74,81,75,83,85);(56,58)(57,62)(59,70)(60,69)(61,67)(63,66)(64,78)(65,75)(68,72)(71,83)(73,81)(74,76)(77,80)(79,87)(82,84)(85,86);(58,67)(62,66)(68,70)(69,81)(74,78)(75,80)(82,83)(85,87)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26324);