Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13038 |
474 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 107 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)(58,60,65,77)(59,67,71,85)(61,79,63,74)(62,75,81,73)(64,82,80,76)(66,83,69,87)(68,86,78,72)(70,89,88,84);(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)(55,57)(59,66,71,69)(61,62)(63,81)(64,80)(67,83,85,87)(68,70,78,88)(72,84,86,89)(73,74)(75,79)(76,82);(54,55,56,57)(58,59,63,68,64,70,81,69,65,71,61,78,80,88,62,66)(60,72,75,85,76,87,79,89,77,86,73,67,82,83,74,84)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh,i^c=i^52872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86804);