Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊#64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23×C3 |
| unambiguous semidirect product | C3⋊#10593111 |
| derived | C3×C22⋊(C22⋊C9) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#64188 |
| commutator, hypocenter | C3×C22⋊(C22⋊C9) | C4 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting, socle, prime radical | C25×C32 | D3 |
| cosocle | C2×C3×C22⋊(C22⋊C9) | C2 |
| LCS | G ▷ C3×C22⋊(C22⋊C9) | |
| UCS | C1 ◁ C2 | |
866 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^3,2^4|a^2=b,c^3=e,c^a=c^2e^2,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,f^c=g,g^c=fg,h^c=i,i^c=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30622);