Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of A4×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| direct product | C3×A4×C3⋊D3 |
| unambiguous semidirect product | C3⋊(C3×D3×A4) |
| Fitting, socle | C22×C33 ⋊ C2×C3 |
| derived | C22×C32 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×C3⋊D3 |
| commutator, cosocle, hypocenter | C22×C32 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C2×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C3 | |
376 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C3×A4⋊#157, C3⋊(A4⋊#157), C3×A4×#157, … |
| maximal normal subgroup of | C3×A4⋊#157, C3⋊(A4⋊#157), C3×A4×#157, … |
| maximal quotient of | C3×C3⋊(C3⋊(#41⋊C3)), C3×C3⋊D3×SL(2,𝔽3), C3×A4×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 740);