Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C210 |
| direct product | C2×C53×#2883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| prime radical | C22×H(5,𝔽2)×C53 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×D4×C53 ◁ G | |
5338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 5 generators on 71 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)(58,64,66,61)(59,62)(60,63)(65,70,71,68)(67,69);(56,58)(57,61)(59,65)(60,66)(62,68)(63,64)(67,71)(69,70);(56,59)(57,62)(58,65)(60,67)(61,68)(63,69)(64,70)(66,71);(59,67)(62,69)(65,71)(68,70)4346 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53056);