Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×AGL(1,𝔽9) |
| Fitting, socle | C22×C33 ⋊ C2×C8 |
| derived, Sylow | C33 ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C23×C8 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C2×C8 |
| prime radical | C22×D3×C3⋊D3 | C4 |
| cosocle | C3×#157 | C24 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
842 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 228 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D3×AΓL(1,𝔽9) |
| maximal normal subgroup of | C22×D3×AΓL(1,𝔽9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47882);