Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C3×C8×C9) |
| derived | C3 ⋊ C3×C8×C9 |
| Sylow | C32×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4×C9 | D3 |
| commutator, hypocenter | C3 | C3×C8×C9 |
| Frattini | C3×C4 | C32×D3 |
| Fitting | C32×C4×C9 | C2 |
| socle, prime radical | C2×C33 | C3×C4 |
| cosocle | C32×C4 | C2×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C4×C9 | |
125 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D9×C3⋊C8, C3×(C3⋊D9)⋊C8, C3×#88413, … |
| maximal normal subgroup of | C3×D9×C3⋊C8, C3×(C3⋊D9)⋊C8, C3×#88413, … |
| maximal quotient of | C3×C9×C3⋊C16, C2×C3×C9×C3⋊C8, C3×C9×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 233);