Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13097 |
650 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 99 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:
(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)(54,57,59,66)(55,61,64,79)(56,67,70,68)(58,74,75,72)(60,76,78,82)(62,80,71,63)(65,77,83,81)(69,85,73,84);(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)(54,56,66,68,59,70,57,67)(55,60,71,83,64,78,62,65)(58,73,74,84,75,69,72,85)(61,76,63,81,79,82,80,77);(54,55)(56,65,70,83)(57,71,66,62)(58,63)(59,64)(60,68)(61,72,79,74)(67,78)(69,82,73,76)(75,80)(77,85)(81,84)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^52,d^c=dh,e^c=eh,f^c=fh,f^d=fh788 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 87132);