Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (#126⋊C3)⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (#126⋊C3)⋊(C3⋊D3) |
| Fitting | C32×#126 ⋊ C2×C3 |
| derived | C32×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C42 | C22×C3 |
| Frattini | C22 | C2×A4×C3⋊D3 |
| Fitting | C32×#126 | C2×C3 |
| socle | C22×C32 | C22×A4 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 | |
450 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3,2^2,3^2,2^2|d^2=h,e^2=hi,d^a=dh,e^a=ehi,f^a=f^2,g^a=g^2,d^b=dhi,e^b=ei,d^c=de,e^c=dhi,h^c=i,i^c=hi57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39217);