Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C7×C22⋊C4 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #99 |
| Sylow | C7 ⋊ C3×#806 |
242 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^2,3,2^3,3,7|a^2=e,b^a=bd,d^a=df,e^b=ef,g^b=g^2,h^b=h^6,h^c=h^493 conjugacy classes — view the full character table.
| maximal subgroup of | #99×C3⋊AGL(1,𝔽7), D3×#6380⋊C3, AGL(1,𝔽7)×#806, … |
| maximal normal subgroup of | #99×C3⋊AGL(1,𝔽7), D3×#6380⋊C3, AGL(1,𝔽7)×#806, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 3, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);