Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Sylow | C53 ⋊ #36698 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D53 |
| commutator, socle, cosocle | C23×C53 | C25 |
| Fitting | C53×#2820 | C2 |
| prime radical | C22×D4×C53 | C23 |
| hypercenter | #2820 | D53 |
| hypocenter | C53 | #36698 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #2820 | |
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:
(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)(54,56,57,60)(55,59,58,61)(70,74,75,73)(71,77,78,83)(72,80,82,81)(76,85,79,84);(62,64)(63,67)(65,68)(66,69)(70,73,75,74)(71,77,78,83)(72,81,82,80)(76,85,79,84);(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)(55,58)(59,61)(63,66)(64,68)(70,73,75,74)(71,77,78,83)(72,80,82,81)(76,84,79,85);(62,63,65,66)(64,67,68,69)(70,72,75,82)(71,76,78,79)(73,81,74,80)(77,85,83,84);(54,55)(56,59)(57,58)(60,61)(64,68)(67,69)(70,71,75,78)(72,79,82,76)(73,83,74,77)(80,84,81,85)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh,i^c=i^52,i^e=i^52728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 376255);