Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24×C3 |
| direct product | D4×A4×C3⋊D3 |
| Fitting | C22×C32×D4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×C3⋊D3 |
| commutator, socle, cosocle | C23×C32 | C23×C3 |
| Fitting | C22×C32×D4 | C2×C3 |
| hypercenter | D4 | A4×C3⋊D3 |
| hypocenter | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
2030 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 2030 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | A4×C3⋊D3×S4, D4×C3⋊D3×S4, D32×D4×A4, … |
| maximal normal subgroup of | D4×C3⋊D3×S4, D32×D4×A4, D4×#89101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47523);