Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C22⋊(C22⋊A4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C29×C3 |
| direct product | C3×C22⋊(C22⋊(C22⋊C9)) |
| unambiguous semidirect product | (C22×C3)⋊(C22⋊(C22⋊C9)) |
| derived, Sylow | C26 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C22⋊(C22⋊A4) |
| commutator, hypocenter | C26 | C3×C9 |
| Frattini | C3 | C3×C22⋊(C22⋊A4) |
| Fitting, socle, prime radical | C26×C32 | C3 |
| cosocle | C26×C3 | C32 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C32 | |
6002 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, 46178);