Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C27 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×A4×C27 |
| Fitting | C23×C27 ⋊ C3 |
| derived | C22 ⋊ C2×C3×C27 |
| Sylow | C23 ⋊ C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C27 | A4 |
| commutator, hypocenter | C22 | C2×C3×C27 |
| Frattini | C9 | C2×C3×A4 |
| Fitting | C23×C27 | C3 |
| socle | C23×C3 | C3×C9 |
| prime radical | C2×C3×A4 | C9 |
| cosocle | C22×C9 | C2×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C27 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 162 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C27×SL(2,ℤ/4), A4⋊Dic27, A4×Dic27, … |
| maximal normal subgroup of | C27×SL(2,ℤ/4), A4⋊Dic27, A4×Dic27, … |
| maximal quotient of | C4×A4×C27, C2×SL(2,𝔽3)×C27, C27×#41⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 302);