Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C33 |
| direct product | D4×C3⋊(C3⋊S4) |
| Fitting | C22×C32×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊(C3⋊S4) |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting | C22×C32×D4 | D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | D4 | C3⋊(C3⋊S4) |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
2268 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 2268 conjugacy classes of subgroups — too many to draw. Only its 217 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | S4×C3⋊(C3⋊S4), D4×C3⋊D3×S4, D3×D4×C3⋊S4 |
| maximal normal subgroup of | D4×C3⋊D3×S4, D3×D4×C3⋊S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47460);