Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C9⋊C8)⋊C9 |
| unambiguous semidirect product | (C9⋊C9)⋊C8 |
| derived | C9 ⋊ C8×C9 |
| Sylow | C9⋊C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C8×C9 |
| Frattini | C32×C4 | C3×D3 |
| Fitting | C4×C9⋊C9 | C2 |
| socle, prime radical | C2×C32 | C3×Dic3 |
| cosocle | C3×C4×C9 | C2×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C3×C4 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3,2,3^2|a^2=c,b^3=d,c^2=e,f^3=g,f^a=f^2g^2,g^a=g^2,f^b=fg^2120 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D9⋊C9, #3663⋊C9, C2×(C9⋊C8)⋊C9, … |
| maximal normal subgroup of | C8×D9⋊C9, #3663⋊C9, C2×(C9⋊C8)⋊C9, … |
| maximal quotient of | (C9⋊C9)⋊C16, C2×(C9⋊C8)⋊C9, #14515618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 15);