Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #5450 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | C24 ⋊ C22×C27 |
| Sylow | H(3,𝔽4) ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #5450 |
| commutator, hypocenter | C24 | C22×C27 |
| Frattini | C22×C9 | C22×A4 |
| Fitting | C9×H(3,𝔽4) | C3 |
| socle | C22×C3 | C22×C22⋊C9 |
| prime radical | C3×H(3,𝔽4) | C9 |
| cosocle | C24×C9 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C9 | |
178 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3^3,2^4|c^3=d,d^3=e,f^a=fh,g^a=ghi,f^b=fhi,g^b=gi,f^c=fg,g^c=f,h^c=i,i^c=hi153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2251);