Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #6192⋊C3 |
| Sylow | C3×3−1+2 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C3×C9 | C22×C3 |
| Frattini | C2×C3 | C3×D32 |
| Fitting | C2×C3×3−1+2 | C22 |
| socle | C2×C32 | C2×C3×D3 |
| hypocenter | C3×C9 | C3×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 | |
141 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3,2,3^3|a^2=d,e^3=g,b^a=bd,f^a=f^2,e^b=e^2g^2,g^b=g^2,e^c=eg^251 conjugacy classes — view the full character table.
| maximal subgroup of | #88634⋊C3, #88635⋊C3, #88640⋊C3, … |
| maximal normal subgroup of | #88634⋊C3, #88635⋊C3, #88640⋊C3, … |
| maximal quotient of | #88415⋊C3, #88416⋊C3, #88417⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 382);