Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13026 |
486 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(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)(58,60)(59,65)(61,76)(62,75)(63,80)(64,74)(66,85)(67,84)(68,89)(69,83)(70,88)(71,79)(72,78)(73,77)(81,87)(82,86);(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)(58,60)(59,65)(61,75)(62,76)(63,80)(64,74)(66,84)(67,85)(68,89)(69,83)(70,87)(71,78)(72,79)(73,77)(81,88)(82,86);(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)(54,55,57,56)(58,59,62,67,77,86,78,87,64,69,61,66,63,68,79,88)(60,70,72,82,80,85,75,65,74,81,71,89,73,84,76,83)Polycyclic presentation:
2^8,53|a^2=f,b^2=dfgh,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,i^a=i^52,d^b=dg,e^b=egh,f^b=fg,g^b=gh,i^b=i^52,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86725);