Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C81 · Schur multiplier (H2) = C2 |
| direct product | C2×C22⋊C81 |
| unambiguous semidirect product | C23⋊C81 |
| derived | C22 ⋊ C2×C81 |
| Sylow | C23 ⋊ C81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C27 | A4 |
| commutator, hypocenter | C22 | C2×C81 |
| Frattini | C27 | C2×A4 |
| Fitting | C23×C27 | C3 |
| socle, prime radical | C23×C3 | C27 |
| cosocle | C22×C27 | C2×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C27 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 162 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | (C22⋊C81)⋊C4, C4×C22⋊C81, C2×(C22⋊C81)⋊C2, … |
| maximal normal subgroup of | (C22⋊C81)⋊C4, C4×C22⋊C81, C2×(C22⋊C81)⋊C2, … |
| maximal quotient of | C4×C22⋊C81, C2×Q8⋊C81, #41⋊C81, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 42);