Finite Group DatabasebetaFinite Group Databasebeta| order | 2646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived | C72×C9 ⋊ C2×C3 |
| Hall | #13050515 ⋊ C2 |
| Sylow | C72 ⋊ AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C72×C9 | C2×C3 |
| Frattini | C3 | C3⋊#64423 |
| socle | C3×C72 | C3×D3 |
| LCS | G ▷ C72×C9 | |
| UCS | C1 | |
76 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3^2,7^2,3|c^3=f,c^a=c^2f^2,c^b=cf,d^a=d^6,d^b=d^4,e^a=d^2e,e^b=e^4,f^a=f^285 conjugacy classes — view the full character table.
| maximal subgroup of | #72229923, #72229968, #72229991, … |
| maximal normal subgroup of | #72229923, #72229968, #72229991, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 3, 7, 7, 3]);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 4, 2, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);