Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C22×C32 |
| unambiguous semidirect product | #543⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#543⋊C3)⋊C3 |
| Fitting | C32×#543 ⋊ C3 |
| Sylow | #543 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C3×C22⋊A4 |
| commutator, cosocle | C3×#543 | C32 |
| Fitting | C32×#543 | C3 |
| hypercenter | C22×C32 | C22⋊A4 |
| hypocenter | #543 | H(3,𝔽3) |
| LCS | G ▷ C3×#543 ▷ #543 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C32 | |
413 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 40 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:
3^3,2^6|f^2=h,g^2=h,b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,g^d=gi,f^e=fi,g^f=gh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18321);