Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of AGL(1,𝔽9)⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | AGL(1,𝔽9)⋊S4 |
| Fitting, socle | C22×C32 ⋊ #204 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | A4×#157 | C22 |
| Fitting, socle | C22×C32 | #204 |
| prime radical | #157⋊S4 | C2 |
| hypocenter | C32×A4 | SD16 |
| LCS | G ▷ A4×#157 ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 | |
185 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^4,3^3,2^2|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^2,h^a=hi,f^b=f^2g^2,g^b=fg^2,f^c=g^2,g^c=f,f^d=f^2,g^d=g^2,h^e=i,i^e=hi27 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AΓL(1,𝔽9), C2×AGL(1,𝔽9)⋊S4 |
| maximal normal subgroup of | S4×AΓL(1,𝔽9), C2×AGL(1,𝔽9)⋊S4 |
| maximal quotient of | C2×AGL(1,𝔽9)⋊S4, #72244569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47841);