Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#2123 ⋊ C2 |
| Sylow | C53 ⋊ #12770 |
752 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(2,9)(4,6)(8,22)(10,26)(11,27)(12,18)(13,28)(14,21)(17,19)(23,25)(24,30)(29,31)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,3,7,16)(2,8,12,25)(4,14,6,21)(5,15,20,32)(9,22,18,23)(10,13,11,24)(17,29,19,31)(26,28,27,30)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,6,11,7,12,4,10)(3,13,14,8,16,24,21,25)(5,18,19,27,20,9,17,26)(15,30,29,22,32,28,31,23)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,d^c=dg,f^c=fh,i^c=i^52,e^d=eh,f^d=fh632 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85425);