Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #11993 |
360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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)(55,61)(57,70)(60,74)(62,64)(65,76)(66,83)(69,84)(71,81);(54,56,59,68)(55,60,61,74)(57,69,70,84)(58,72,73,67)(62,76,64,65)(63,79,77,82)(66,71,83,81)(75,78,85,80);(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)(54,55,57,63,58,62,71,78,59,61,70,77,73,64,81,80)(56,65,66,82,67,74,84,85,68,76,83,79,72,60,69,75)Polycyclic presentation:
2^8,53|a^2=e,b^2=h,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,i^a=i^52,d^b=dg,e^b=efgh,f^b=fh,e^c=eh860 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 81104);