Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Sylow | C53 ⋊ #36697 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Fitting | C53×#2823 | C2 |
| prime radical | C22×D4×C53 | C23 |
| hypercenter | #2823 | D53 |
| hypocenter | C53 | #36697 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2823 | |
1484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 85 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)(54,56,57,60)(55,59,58,61);(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)(62,64)(63,67)(65,68)(66,69)(70,73,74,81)(71,76,77,83)(72,79,80,85)(75,82,78,84);(55,58)(59,61)(63,66)(64,68)(73,81)(76,83)(79,85)(82,84);(62,63,65,66)(64,67,68,69)(70,72,74,80)(71,75,77,78)(73,79,81,85)(76,82,83,84);(54,55)(56,59)(57,58)(60,61)(64,68)(67,69)(70,71,74,77)(72,78,80,75)(73,76,81,83)(79,84,85,82)Polycyclic presentation:
2^8,53|a^2=f,b^2=fh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh,i^e=i^52728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 376232);