Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #11994 |
258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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)(57,70)(62,79)(67,83)(69,74)(72,81)(75,84)(77,80)(78,85);(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)(54,56,59,66)(55,60)(57,69,70,74)(58,73,71,68)(61,65)(62,78)(63,82)(64,76)(67,72,83,81)(75,80)(77,84)(79,85);(54,55,57,62,71,63,81,80,59,64,70,79,58,61,72,77)(56,65,67,78,73,76,69,84,66,82,83,85,68,60,74,75)Polycyclic presentation:
2^8,53|a^2=e,b^2=g,d^2=gh,e^2=fg,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efg,f^b=fh,i^b=i^52,e^c=eh860 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 81106);