Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10593735⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3×C4 |
| unambiguous semidirect product | #10593735⋊C3 |
| Fitting, derived | C32×C42 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C32×C42 | C3×C4 |
| Frattini | C22 | A4×#157 |
| socle | C22×C32 | C4×A4 |
| prime radical | C3⋊D3×C42⋊C3 | C2 |
| cosocle | C42×C3⋊D3 | C2×C3 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 | |
160 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 15 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:
2,3,2^3,3^2,2^2|a^2=c,d^2=h,e^2=hi,d^a=dh,e^a=ehi,f^a=fg,g^a=fg^2,d^b=de,e^b=dhi,h^b=i,i^b=hi,f^c=f^2,g^c=g^236 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31904);