Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #14554267 ⋊ C2 |
| derived | #10624733 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(#6912⋊C2) |
| commutator, cosocle, hypocenter | #10624733 | C2×C3 |
| Frattini | C33 | C3×C3⋊(C3⋊D3) |
| Fitting | #14554267 | C2 |
| socle | C32 | C3×C3⋊#272 |
| LCS | G ▷ #10624733 | |
| UCS | C1 ◁ C3 | |
2047 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 2047 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3^7|c^3=h,e^3=h^2,c^a=c^2h^2,d^a=d^2,d^c=df,e^a=e^2h,e^b=eh^2,e^c=ef^2g,g^a=gh^2,g^c=gh,h^a=h^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72222257, #72222315 |
| maximal normal subgroup of | #72222315 |
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, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.8^2);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.6^2*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);