Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#3724 |
| unambiguous semidirect product | (C22×#3724)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D32×A4 |
| commutator | C23×C32 | C2×C3×C4 |
| Fitting, socle, cosocle | C24×C32 | C22×C3 |
| prime radical | C22×C32×A4 | C22 |
| hypocenter | C22×C32 | C3×C4⋊C4 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C22 | |
602 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32231);