Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C3×C9 ⋊ D3 |
| derived | (C3×A4)⋊C9 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C9⋊S4 |
| commutator, cosocle, hypocenter | (C3×A4)⋊C9 | C2 |
| Frattini | C32 | C3⋊S4 |
| Fitting | C22×C3×C9 | D3 |
| socle | C22×C32 | C3⋊D3 |
| LCS | G ▷ (C3×A4)⋊C9 | |
| UCS | C1 ◁ C3 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^4,2^2|c^3=d,b^a=b^2,c^a=c^2d^2,d^a=d^2,f^a=g,g^a=f,c^b=ce,f^b=g,g^b=fg39 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊S4)⋊C9)⋊C2, C2×#10619754, #14517997, … |
| maximal normal subgroup of | ((C3⋊S4)⋊C9)⋊C2, C2×#10619754, #14517997, … |
| maximal quotient of | #13046995, #13046996, #13047014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 273);