Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of A4×C3⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×C3⋊D9 |
| unambiguous semidirect product | C3⋊(A4×D9) |
| Fitting, derived | C22×C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C3×C9 | C2×C3 |
| Frattini | C3 | A4×C3⋊D3 |
| socle | C22×C32 | C3×D3 |
| LCS | G ▷ C22×C3×C9 | |
| UCS | C1 | |
150 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | S4×C3⋊D9, #13049217, D3×A4×D9, … |
| maximal normal subgroup of | S4×C3⋊D9, #13049217, D3×A4×D9, … |
| maximal quotient of | C3⋊#88615, SL(2,𝔽3)×C3⋊D9, A4×C3⋊Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 590);