Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×C3⋊#5934 |
| unambiguous semidirect product | C3⋊(C3×#5934) |
| Fitting | C32×H(5,𝔽2) ⋊ D3 |
| derived | C3×#997 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C3×#997 | C2×C3 |
| Frattini | C2 | C3×C3⋊#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | #1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2×C3 | |
596 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #72245451, C3×D3×#5934 |
| maximal normal subgroup of | C3×D3×#5934 |
| maximal quotient of | #72245451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47488);