Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| direct product | S4×C27 |
| unambiguous semidirect product | A4⋊(C2×C27) |
| Fitting | C22×C27 ⋊ D3 |
| derived | A4 ⋊ C2×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C27 | S4 |
| commutator, hypocenter | A4 | C2×C27 |
| Frattini | C9 | C3×S4 |
| Fitting | C22×C27 | D3 |
| socle | C22×C3 | D3×C9 |
| prime radical | C3×S4 | C9 |
| cosocle | C9×A4 | C2×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C27 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | S4×D27, C2×S4×C27, S4×C81, … |
| maximal normal subgroup of | S4×D27, C2×S4×C27, S4×C81, … |
| maximal quotient of | C27×#226, C27×GL(2,𝔽3), C27×SL(2,ℤ/4), … |
| automorphism group of | C22⋊C81 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 254);