Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #12678 |
292 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 123 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 — 3 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)(55,63,67,66)(57,74)(58,75)(62,77,79,83)(64,81,73,65)(68,78,82,80)(69,84)(70,85);(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)(54,56,59,71,61,72,60,76)(55,62,63,77,67,79,66,83)(57,70,75,84,74,85,58,69)(64,80,65,82,73,78,81,68);(54,55,58,65,60,66,74,64,61,67,75,81,59,63,57,73)(56,68,70,79,71,80,84,83,72,82,85,62,76,78,69,77)Polycyclic presentation:
2^8,53|a^2=e,b^2=f,c^2=de,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=dh,d^b=dg,e^b=egh,g^b=gh,d^c=dh,e^c=eh,i^c=i^52878 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 84994);