Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×A4×AGL(1,𝔽9) |
| unambiguous semidirect product | (C23×AGL(1,𝔽9))⋊C3 |
| Fitting, socle | C23×C32 ⋊ C3×C8 |
| derived | C22×C32 ⋊ C2×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×AGL(1,𝔽9) |
| commutator, hypocenter | C22×C32 | C2×C3×C8 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3×C8 |
| prime radical | C2×A4×C3⋊D3 | C4 |
| cosocle | C22×#157 | C22×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C2 | |
233 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AΓL(1,𝔽9), C2×S4×AGL(1,𝔽9), C2×AGL(1,𝔽9)⋊S4 |
| maximal normal subgroup of | C2×A4×AΓL(1,𝔽9), C2×S4×AGL(1,𝔽9), C2×AGL(1,𝔽9)⋊S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47865);