Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C92 · Schur multiplier (H2) = C2×C9 |
| direct product | C2×C9×C22⋊C9 |
| unambiguous semidirect product | C23⋊C92 |
| derived | C22 ⋊ C2×C92 |
| Sylow | C23 ⋊ C92 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C9 | A4 |
| commutator, hypocenter | C22 | C2×C92 |
| Frattini | C32 | C2×C3×A4 |
| Fitting | C23×C3×C9 | C3 |
| socle, prime radical | C23×C32 | C32 |
| cosocle | C22×C32 | C2×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C3×C9 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 162 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C9×(C22⋊C9)⋊C4, C9⋊((C22⋊C9)⋊C4), Dic9×C22⋊C9, … |
| maximal normal subgroup of | C9×(C22⋊C9)⋊C4, C9⋊((C22⋊C9)⋊C4), Dic9×C22⋊C9, … |
| maximal quotient of | C4×C9×C22⋊C9, C2×C9×Q8⋊C9, C9×#41⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 306);