Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4⋊AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×A4⋊AGL(1,𝔽9) |
| unambiguous semidirect product | (C2×A4)⋊AGL(1,𝔽9) |
| Fitting, socle | C23×C32 ⋊ C3⋊C8 |
| derived | C32×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊AGL(1,𝔽9) |
| commutator, hypocenter | C32×A4 | C2×C8 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C3⋊C8 |
| prime radical | C2×A4×C3⋊D3 | C4 |
| cosocle | A4×#157 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
225 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13181780, C2×S4×AGL(1,𝔽9), C2×#13181779 |
| maximal normal subgroup of | C2×#13181780, C2×S4×AGL(1,𝔽9), C2×#13181779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47864);