Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C53×#510 ⋊ C2 |
| Sylow | C53 ⋊ #13342 |
558 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 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)(55,57)(58,61)(59,68)(60,62)(63,76)(64,75)(65,81)(66,74)(67,83)(69,79)(70,86)(71,87)(72,84)(73,88)(77,80)(78,82)(85,89);(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)(54,55)(56,57)(58,60)(59,67)(61,80)(62,81)(63,74)(64,77)(65,76)(66,78)(68,70)(69,87)(71,86)(72,85)(73,79)(75,82)(83,84)(88,89);(58,59,66,73)(60,68,78,88)(61,79,74,67)(62,70,82,89)(63,69,80,83)(64,72,65,71)(75,86,81,85)(76,87,77,84)Polycyclic presentation:
2^8,53|a^2=h,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=egh,f^c=fh,g^c=gh,i^c=i^52866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88500);