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Cayley diagram

C3×C3⋊AGL(1,ℤ/9) #92387

order486
isoclinismisoclinic class of C3⋊AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productC3×C3⋊AGL(1,ℤ/9)
unambiguous semidirect productC3⋊(C3×AGL(1,ℤ/9))

Splittings

Fitting, SylowC32×3−1+2 ⋊ C2
derivedC3×C9 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC3C3⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC3×C9C2×C32
FrattiniC3C32×C3⋊D3
FittingC32×3−1+2C2
socleC33C3×D3
LCSG ▷ C3×C9
UCSC1 ◁ C3

234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×AGL(1,ℤ/9), D3×C3⋊AGL(1,ℤ/9), #66431⋊C3, …
maximal normal subgroup ofC3×D3×AGL(1,ℤ/9), D3×C3⋊AGL(1,ℤ/9), #66431⋊C3, …
maximal quotient ofC3×C3⋊(3−1+2⋊C4), C2×C3×C3⋊AGL(1,ℤ/9), C3×C3⋊#4169, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 232);