Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C42×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ #560 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D3×D9 |
| commutator, cosocle | C22×C3×C9 | C24 |
| Frattini | C22×C3 | C22×D32 |
| Fitting | C3×C42×C9 | C22 |
| socle | C22×C32 | C23×D3 |
| hypercenter | C42 | D3×D9 |
| hypocenter | C3×C9 | #560 |
| LCS | G ▷ C22×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C42 | |
864 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|c^2=ef,d^2=e,g^3=i,c^a=cf,d^a=def,g^a=g^2i^2,i^a=i^2,c^b=ce,d^b=df,h^b=h^2153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13812);