Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C23×C3×C4 |
| direct product | C3×C22⋊(C42⋊C9) |
| unambiguous semidirect product | (C22×C3)⋊(C42⋊C9) |
| derived, Sylow | C22×C42 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C22⋊(C42⋊C3) |
| commutator, hypocenter | C22×C42 | C3×C9 |
| Frattini | C22×C3 | C3×C22⋊A4 |
| Fitting | C22×C32×C42 | C3 |
| socle, prime radical | C24×C32 | A4 |
| cosocle | C22×C3×C42 | C32 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C32 | |
594 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18316);