Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(#5934⋊C3) |
| Fitting | C3×H(5,𝔽2) ⋊ C3×D3 |
| derived | C3×#997 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(#1020⋊C3) |
| commutator, cosocle, hypocenter | C3×#997 | C2×C3 |
| Fitting | C3×H(5,𝔽2) | C3×D3 |
| socle | C2×C3 | #1020⋊C3 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
230 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^3,2^5|e^2=i,f^2=i,g^2=i,h^2=i,c^a=c^2,d^a=d^2,e^a=gi,f^a=hi,g^a=ei,h^a=fi,e^b=f,f^b=efi,g^b=h,h^b=gh,e^c=f,f^c=efi,g^c=gh,h^c=g,f^e=fi,h^g=hi48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46394);