Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12865 |
334 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 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)(54,57,58,71,59,72,73,85)(55,61,62,77,64,78,79,83)(56,66,67,81,68,70,82,84)(60,74,65,80,75,76,63,69);(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)(55,60,64,75)(57,70)(58,67)(59,68)(61,76,78,74)(62,65,79,63)(66,72)(69,83,80,77)(71,84)(73,82)(81,85);(54,55)(56,65,68,63)(57,69)(58,62)(59,64)(60,67,75,82)(61,66,78,70)(71,74)(72,80)(73,79)(76,85)(77,81,83,84)Polycyclic presentation:
2^8,53|a^2=f,b^2=g,c^2=d,d^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,f^b=fh,i^b=i^52,f^c=fg,i^c=i^52,f^e=fh710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85851);