Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #7433 |
182 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 generators on 85 points:
(1,3,13,15,6,17,5,16)(2,7,9,23,11,24,10,27)(4,19,20,31,22,32,21,14)(8,25,28,12,26,29,18,30)(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);(1,2,5,9,6,11,13,10)(3,12,15,29,17,30,16,25)(4,18,20,26,22,28,21,8)(7,19,23,31,24,32,27,14)(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)Polycyclic presentation:
2^8,53|a^2=d,b^2=eg,c^2=fgh,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defgh,e^b=eh,g^b=gh,i^b=i^52,d^c=dgh,e^c=eh1019 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57167);