Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#11724113 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3×C3⋊D3 | C24 |
| Frattini | C2 | C2×D3×GO+(4,𝔽2) |
| Fitting, socle | C22×C33 | C2×D4 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C22 | |
2496 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 2496 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46611);