Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Sylow | C53 ⋊ C4×#564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D53 |
| commutator | C22×C53 | C24×C4 |
| Fitting | C53×#2382 | C2 |
| socle, prime radical, cosocle | C23×C53 | C25 |
| hypercenter | #2382 | D53 |
| hypocenter | C53 | C4×#564 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2382 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 5 generators on 89 points:
(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)(54,56,57,55);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(59,65)(63,72)(64,68)(71,73)(74,77,78,87)(75,80,81,85)(76,84,83,86)(79,88,82,89);(58,61,62,70)(59,64,65,68)(60,66,67,69)(63,71,72,73)(76,83)(79,82)(84,86)(88,89);(58,60,62,67)(59,63,65,72)(61,69,70,66)(64,73,68,71)(74,76)(75,79)(77,86)(78,83)(80,89)(81,82)(84,87)(85,88);(54,55,57,56)(58,59)(60,63)(61,68)(62,65)(64,70)(66,73)(67,72)(69,71)(74,75,78,81)(76,82,83,79)(77,85,87,80)(84,88,86,89)Polycyclic presentation:
2^8,53|a^2=h,b^2=g,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,i^e=i^521064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 332288);