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Cayley diagram

C3×AGL(1,ℤ/9) #4200

order162
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = D3×AGL(1,ℤ/9) (order 324) · Inn(G) = AGL(1,ℤ/9) · Out(G) = D3
holomorphG ⋊ Aut(G) = #72269259
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊C32
unambiguous semidirect product(C3×D9)⋊C3

Splittings

Fitting, SylowC3×3−1+2 ⋊ C2
derivedC9 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C2×C32
FrattiniC3C32×D3
FittingC3×3−1+2C2
socleC32C3×D3
LCSG ▷ C9
UCSC1 ◁ C3

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 2 generators on 12 points:

(5,9,12,7,8,11)(6,10);(1,2,3)(4,12,7,10,8,9,6,5,11)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,ℤ/9), C2×C3×AGL(1,ℤ/9), C9×AGL(1,ℤ/9), …
maximal normal subgroup ofD3×AGL(1,ℤ/9), C2×C3×AGL(1,ℤ/9), C9×AGL(1,ℤ/9), …
maximal quotient ofC3×3−1+2⋊C4, C2×C3×AGL(1,ℤ/9), C3×#4169, …
automorphism group ofD9⋊C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 36);