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Cayley diagram

C9×AGL(1,ℤ/9) #92255

order486
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C3×D3×AGL(1,ℤ/9) (order 972) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C3×D3
homologyAbelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3

Aliases

direct productC9×AGL(1,ℤ/9)

Splittings

Fitting, SylowC9×3−1+2 ⋊ C2
derivedC9 ⋊ C2×C3×C9

Subgroups

subgroupG/H
center, hypercenterC9AGL(1,ℤ/9)
commutator, hypocenterC9C2×C3×C9
Frattini, socleC32C32×D3
FittingC9×3−1+2C2
prime radicalC3×AGL(1,ℤ/9)C3
cosocleC3×C9C2×C32
LCSG ▷ C9
UCSC1 ◁ C9

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 18 points:

(11,15,18,13,14,17)(12,16);(1,2,4,3,5,7,6,8,9)(10,18,13,16,14,15,12,11,17)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×AGL(1,ℤ/9), C2×C9×AGL(1,ℤ/9), C9×#4183, …
maximal normal subgroup ofD9×AGL(1,ℤ/9), C2×C9×AGL(1,ℤ/9), C27×AGL(1,ℤ/9), …
maximal quotient ofC9×3−1+2⋊C4, C2×C9×AGL(1,ℤ/9), C9×#4169, …
automorphism group ofD9⋊C27

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 100);