Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | #578⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#578⋊C3)⋊C3 |
| Fitting | C32×#578 ⋊ C3 |
| Sylow | #578 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×D4×A4 |
| commutator, cosocle | C3×#41 | C22×C32 |
| Frattini | C3×C4 | C22×C3×A4 |
| Fitting | C32×#578 | C3 |
| hypercenter | C32×D8 | A4 |
| hypocenter | Q8 | D4×H(3,𝔽3) |
| LCS | G ▷ C3×#41 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C32×C4 ◁ C32×D8 | |
419 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 63 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:
2^2,3^2,2,3,2^3|e^2=i,g^2=i,h^2=i,b^a=be,e^a=ei,e^b=ei,d^c=df,g^c=gh,h^c=g,h^g=hi122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18303);