Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Fitting | C32×H(3,𝔽4) ⋊ C3 |
| Sylow | H(3,𝔽4) ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#5450 |
| commutator, cosocle | C24×C3 | C22×C32 |
| Frattini, socle | C22×C3 | C22×C3×A4 |
| Fitting | C32×H(3,𝔽4) | C3 |
| hypercenter | C32 | #5450 |
| hypocenter | C24 | C22×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C32 | |
404 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 53 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:
2^2,3^3,2^4|f^a=fh,g^a=ghi,f^b=fhi,g^b=gi,d^c=de,f^c=fg,g^c=f,h^c=i,i^c=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18273);