Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C3⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C33 |
| direct product | C22×C3×C3⋊(C3⋊D3) |
| Fitting, socle | C22×C34 ⋊ C2 |
| derived | C33 ⋊ C23×C3 |
| Sylow | C34 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | C3⋊(C3⋊D3) |
| commutator, cosocle, hypocenter | C33 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22×C3 | |
1286 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#88804, #13049061, #13049094, … |
| maximal normal subgroup of | C3×#88804, #13049061, #13049094, … |
| maximal quotient of | C2×C3×#6308, C2×C3×C4×C3⋊(C3⋊D3), C2×C3×C4⋊(C3⋊(C3⋊D3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 755);