Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | A4×#3712 |
| unambiguous semidirect product | (C22×#3712)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32×A4 |
| commutator, socle | C23×C32 | C2×C3×C4 |
| Fitting, cosocle | C22×C32×C4 | C22×C3 |
| prime radical | C2×A4×C3⋊D3 | C4 |
| hypocenter | C22×C32 | C3×M16 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C4 | |
442 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32190);