Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C32 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×#63725 |
| derived | C32 ⋊ C2×C3×C32 |
| Sylow | C33 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C2×C3×C32 |
| Frattini | C4 | C2×C3×AGL(1,𝔽9) |
| Fitting | C2×C33×C4 | C8 |
| socle, prime radical | C22×C33 | C16 |
| cosocle | #3708 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3×C4 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32248);