Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C32×D3×A4 |
| unambiguous semidirect product | C3⋊(C2×C32×A4) |
| unambiguous semidirect product | (C22×D3)⋊C33 |
| unambiguous semidirect product | (C22×C3×D3)⋊C32 |
| Fitting, socle | C22×C33 ⋊ C2×C3 |
| derived | C22×C3 ⋊ C2×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D3×A4 |
| commutator, cosocle, hypocenter | C22×C3 | C2×C33 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C2×C3 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C32 | |
436 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D3×S4, C3×D3×C3⋊S4, D3×C3⋊(C3⋊S4), … |
| maximal normal subgroup of | C32×D3×S4, C3×D3×C3⋊S4, D3×C3⋊(C3⋊S4), … |
| maximal quotient of | C32×C3⋊(#41⋊C3), C32×D3×SL(2,𝔽3), C32×Dic3×A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 739);