Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×#120⋊C9 |
| unambiguous semidirect product | (C2×C3×#120)⋊C9 |
| derived | C24 ⋊ C22×C3×C9 |
| Sylow | C2×#120 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | #120⋊C3 |
| commutator, hypocenter | C24 | C22×C3×C9 |
| Frattini | C22×C3 | C22×C3×A4 |
| Fitting, prime radical | C2×C32×#120 | C3 |
| socle | C23×C32 | C2×A4 |
| cosocle | C24×C3 | C22×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C32 | |
814 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18244);