Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×C3⋊(#401⋊C3) |
| unambiguous semidirect product | C3⋊(C3×#401⋊C3) |
| unambiguous semidirect product | (C3⋊#401)⋊C32 |
| unambiguous semidirect product | (C3×C3⋊#401)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(#126⋊C3) |
| commutator, cosocle, hypocenter | C3×#95 | C2×C32 |
| Frattini | C23 | C3×D3×A4 |
| Fitting | C32×#95 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| prime radical | C32×#95⋊C3 | C2 |
| LCS | G ▷ C3×#95 | |
| UCS | C1 ◁ C2×C3 | |
215 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23088);