Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (#41⋊C3)⋊(C22⋊C9) |
| unambiguous semidirect product | #5945⋊C9 |
| Sylow | #585 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | A42 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C9 |
| Frattini, socle | C2×C3 | C2×A42 |
| Fitting | C3×#585 | C32 |
| prime radical | C3×#5945 | C3 |
| cosocle | C3×H(5,𝔽2) | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C3×C4 | |
261 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,c^3=d,e^2=i,f^2=i,g^2=i,h^2=i,g^b=gh,h^b=g,e^c=ef,f^c=e,f^e=fi,h^g=hi150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30351);