Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #13056 |
632 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 632 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,8,11,10)(6,13,17,32)(7,19,22,20)(9,25,26,23)(12,27,30,36)(14,15,16,31)(18,33,29,28)(21,35,24,34)(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)(86,87)(88,89);(1,3,4,2)(5,7)(6,12)(8,19)(9,24)(10,20)(11,22)(13,27)(14,29)(15,28)(16,18)(17,30)(21,26)(23,35)(25,34)(31,33)(32,36)(37,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39);(1,2,4,3)(5,6,10,16,11,17,8,14)(7,18,19,12,22,29,20,30)(9,15,23,32,26,31,25,13)(21,27,35,33,24,36,34,28)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,d^b=dh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,i^c=i^52,e^d=eh,f^e=fh866 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86912);