Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#2239 ⋊ C2 |
| Sylow | C53 ⋊ #33690 |
526 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 4 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)(57,68,69,65)(58,74)(62,76,77,78)(63,81)(66,72)(67,80,84,82)(70,73,83,85)(71,79);(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)(54,57)(55,62)(56,65)(58,73)(59,68)(60,69)(61,76)(63,80)(64,77)(66,83)(67,71)(70,72)(74,85)(75,78)(79,84)(81,82);(54,56,60,59)(55,61,64,75)(57,65,69,68)(58,72,74,66)(62,76,77,78)(63,79,81,71)(67,80,84,82)(70,85,83,73);(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)(56,61)(57,67)(58,71)(59,75)(60,64)(62,70)(63,72)(65,82)(66,81)(68,80)(69,84)(73,76)(74,79)(77,83)(78,85)Polycyclic presentation:
2^8,53|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,i^c=i^52,f^d=fh,i^d=i^52554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 324819);