Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×(C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
| direct product | Dic3×(C22⋊C9)⋊C4 |
| derived | C3×C22⋊C9 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×(C22⋊C9)⋊C2 |
| commutator, hypocenter | C3×C22⋊C9 | C42 |
| Frattini | C22×C3 | D3×S4 |
| Fitting, socle, prime radical | C24×C32 | C2×D3 |
| cosocle | C22×C3×C22⋊C9 | C22 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C22 | |
503 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12454);