Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C2×C33 |
| direct product | C33×C4×D3 |
| Fitting | C34×C4 ⋊ C2 |
| derived | C3 ⋊ C2×C33×C4 |
| Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C33×C4 | D3 |
| commutator, hypocenter | C3 | C2×C33×C4 |
| Frattini | C2 | C2×C33×D3 |
| Fitting | C34×C4 | C2 |
| socle | C2×C34 | C22 |
| prime radical | C2×C33×D3 | C2 |
| cosocle | C2×C3 | C22×C33 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C33×C4 | |
682 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×D3×C3⋊C8, C32×#3711, C3×D3×C3⋊(C3⋊C8), … |
| maximal normal subgroup of | C32×D3×C3⋊C8, C32×#3711, C3×D3×C3⋊(C3⋊C8), … |
| maximal quotient of | C33×D3×C8, C33×ΓL(1,𝔽25), C33×C4×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 680);