Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #12306 |
274 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,3)(2,9,12,13)(4,19)(5,16)(6,15)(7,23)(8,24,17,28)(10,25,27,29)(11,26,22,30)(14,20)(18,32)(21,31)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(2,8)(4,18)(5,7)(9,24)(10,22)(11,27)(12,17)(13,28)(14,31)(16,23)(19,32)(20,21)(25,30)(26,29)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,2,7,11,6,12,5,22)(3,13,23,30,15,9,16,26)(4,17,18,27,20,8,21,10)(14,24,31,25,19,28,32,29)Polycyclic presentation:
2^8,53|a^2=fg,c^2=gh,d^2=f,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,i^c=i^52,e^d=eh545 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 82935);