Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D3×C32 |
| Fitting | C33×C32 ⋊ C2 |
| derived | C3 ⋊ C2×C32×C32 |
| Sylow | C33 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C32 | D3 |
| commutator, hypocenter | C3 | C2×C32×C32 |
| Frattini | C16 | C2×C32×D3 |
| Fitting | C33×C32 | C2 |
| socle | C2×C33 | C2×C16 |
| prime radical | C2×C32×D3 | C16 |
| cosocle | C3×C16 | C22×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C32×C32 | |
252 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
864 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×C32×(C3⋊D3)⋊C3, C3×D32×C32, C3×C32×#272 |
| maximal normal subgroup of | C3×D32×C32 |
| maximal quotient of | C3×C32×(C3⋊D3)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11811);