Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#3774 |
| unambiguous semidirect product | (C22×#3774)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×GO+(4,𝔽2) |
| commutator, cosocle | C22×C3⋊Dic3 | C22×C3 |
| Fitting, socle | C23×C32 | C3×D4 |
| hypocenter | C22×C32 | C3×D8 |
| LCS | G ▷ C22×C3⋊Dic3 ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
358 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46291);