Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#3713 |
| unambiguous semidirect product | (C22×#3713)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×#625 |
| commutator, Fitting, cosocle | C22×C32×C4 | C22×C3 |
| Frattini, hypercenter | C4 | D32×A4 |
| socle | C23×C32 | C3×D4 |
| hypocenter | C22×C32 | C3×D8 |
| LCS | G ▷ C22×C32×C4 ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
474 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32191);