Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×C3⋊(#120⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | D3×C3⋊(#120⋊C3) |
| unambiguous semidirect product | (C3×D3)⋊(#120⋊C3) |
| unambiguous semidirect product | (D3×#953)⋊C3 |
| Fitting, derived | C24×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C24×C32 | C22×C3 |
| Frattini | C22 | D32×A4 |
| socle | C22×C32 | C22×A4 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
661 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 661 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) = 48 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×C3⋊(#120⋊C3), #72220699, D32×#120⋊C3, … |
| maximal normal subgroup of | C3×D3×C3⋊(#120⋊C3), D32×#120⋊C3, D3×#10593244 |
| maximal quotient of | C3×D3×C3⋊(#120⋊C3), #72220699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32232);