Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#1415 ⋊ C2 |
| Sylow | C53 ⋊ #13654 |
212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 6784 |
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)(54,56,57,55)(59,65)(64,77)(66,80)(67,81)(68,76)(75,89)(78,83)(79,85);(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)(59,64)(60,69)(62,70)(63,74)(65,77)(66,67)(68,85)(71,82)(72,88)(73,84)(75,78)(76,79)(80,81)(83,89)(86,87);(54,55,57,56)(58,59,61,65)(60,64,71,77)(62,75,73,89)(63,76,74,68)(66,82,80,69)(67,84,81,70)(72,85,86,79)(78,88,83,87)Polycyclic presentation:
2^8,53|a^2=e,c^2=eh,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,i^c=i^521040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90168);