Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#361 ⋊ C2 |
| Sylow | C53 ⋊ #13007 |
492 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 109 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)(55,63)(56,68)(58,74)(59,77)(60,79)(61,71)(62,73)(64,83)(65,84)(66,85)(67,82)(69,80)(70,76)(72,75)(78,81);(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)(54,56,59,67,62,65,61,55)(57,68,77,82,73,84,71,63)(58,69,60,78,76,64,75,66)(70,83,72,85,74,80,79,81);(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)(54,55,60,66,59,65,76,64,62,67,75,78,61,56,58,69)(57,68,72,83,77,63,74,81,73,84,79,80,71,82,70,85)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfgh,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,i^a=i^52,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,d^c=dh,e^c=eh,f^c=fh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86599);