Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #33691 |
524 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 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)(55,61)(56,65,66,67)(57,71)(60,74,75,79)(63,76)(64,70,81,82)(68,84)(73,85,80,78);(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)(54,56)(55,60)(57,70)(58,65)(59,66)(61,75)(62,78)(63,79)(64,68)(67,72)(69,85)(71,82)(73,77)(74,76)(80,83)(81,84);(54,55)(56,60)(57,69)(58,63)(59,61)(62,71)(64,80)(65,79)(66,75)(67,74)(68,83)(70,85)(72,76)(73,81)(77,84)(78,82);(56,64)(57,68)(58,72)(60,73)(62,77)(63,76)(65,70)(66,81)(67,82)(69,83)(71,84)(74,85)(75,80)(78,79)Polycyclic presentation:
2^8,53|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=dh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh,i^d=i^52554 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 324822);