Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| direct product | S4×AGL(1,𝔽9) |
| Fitting, socle | C22×C32 ⋊ D3×C8 |
| derived | C32×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C32×A4 | C2×C8 |
| Fitting, socle | C22×C32 | D3×C8 |
| prime radical | C3⋊D3×S4 | C4 |
| cosocle | A4×#157 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
199 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
45 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AΓL(1,𝔽9), C2×S4×AGL(1,𝔽9) |
| maximal normal subgroup of | S4×AΓL(1,𝔽9), C2×S4×AGL(1,𝔽9) |
| maximal quotient of | C2×S4×AGL(1,𝔽9), GL(2,𝔽3)×AGL(1,𝔽9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47840);