Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#1418 ⋊ C2 |
| Sylow | C53 ⋊ #13658 |
194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(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)(54,56,67,73,60,58,59,66)(55,62,65,64,63,74,75,81)(57,69,84,83,72,68,71,85)(61,76,79,70,77,80,82,78);(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)(56,62)(57,70)(58,74)(59,75)(60,63)(61,68)(64,73)(65,67)(66,81)(69,77)(71,76)(72,78)(79,85)(80,84)(82,83);(55,61,63,77)(56,66)(57,69)(58,73)(59,67)(62,70,74,78)(64,76,81,80)(65,82,75,79)(68,72)(71,83)(84,85)Polycyclic presentation:
2^8,53|a^2=e,c^2=egh,d^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fg,g^a=gh,d^b=dfh,e^b=eh,f^b=fg,g^b=gh,d^c=dh,i^c=i^52710 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90190);