Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #32876 |
584 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(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)(65,81)(69,84)(75,80)(77,82);(54,57,58,71,59,70,74,85)(55,61,62,78,63,68,72,83)(56,65,67,69,66,81,73,84)(60,75,76,77,64,80,79,82);(54,56,58,73,59,66,74,67)(55,60,62,79,63,64,72,76)(57,69,71,65,70,84,85,81)(61,77,78,75,68,82,83,80);(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)(54,55,59,63)(56,64,66,60)(57,68,70,61)(58,72,74,62)(65,80,81,75)(67,76,73,79)(69,77,84,82)(71,78,85,83)Polycyclic presentation:
2^8,53|a^2=h,b^2=eg,c^2=g,g^2=h,b^a=be,c^a=cf,g^a=gh,i^a=i^52,c^b=cg,d^b=def,e^b=eh,f^b=fh,g^b=gh,d^c=de,f^c=fh704 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 318192);