Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C33 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | C3⋊AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C4×C16 |
| Frattini | C4 | C2×C3⋊AGL(1,𝔽9) |
| Fitting | C33×C8 | C8 |
| socle | C2×C33 | C4×C8 |
| prime radical | C2×C3×C3⋊D3 | C2×C8 |
| cosocle | C3×#3787 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C8 | |
146 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=c,b^2=d,c^2=e,d^2=f,e^2=f,g^a=g^2,h^a=h^2i^2,i^a=h^2,h^c=h^2i,i^c=hi,h^e=h^2,i^e=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32271);