Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(#127⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | C3⋊(#384⋊C9) |
| unambiguous semidirect product | #4935⋊C9 |
| derived | C3×C42 ⋊ C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(#127⋊C3) |
| commutator, hypocenter | C3×C42 | C4×C9 |
| Frattini | C23×C3 | D3×A4 |
| Fitting, cosocle | C2×C32×C42 | C2×C3 |
| socle, prime radical | C23×C32 | C2×A4 |
| LCS | G ▷ C3×C42 | |
| UCS | C1 ◁ C2×C3 | |
194 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 33 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:
2,3,2,3,2^2,3,2^2|a^2=c,b^3=d,e^2=h,f^2=hi,e^a=eh,f^a=fhi,g^a=g^2,e^b=ef,f^b=ehi,h^b=i,i^b=hi108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2906);