Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #13147227 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| 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, socle | C22×C3 | C2×D3×A4 |
| Fitting | C32×#126 | C2×C3 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 | |
339 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 28 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:
2^2,3,2^2,3^2,2^2|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,f^c=fg,h^c=i,i^c=hi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13290);