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Cayley diagram

C2×C3⋊AGL(1,ℤ/9) #66454

order324
isoclinismisoclinic class of C3⋊AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊AGL(1,ℤ/9)
unambiguous semidirect product(C2×C3)⋊AGL(1,ℤ/9)

Splittings

FittingC2×C3×3−1+2 ⋊ C2
derivedC3×C9 ⋊ C22×C3
SylowC3×3−1+2 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2C3⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC3×C9C22×C3
FrattiniC3C2×C3×C3⋊D3
FittingC2×C3×3−1+2C2
socleC2×C32C3×D3
LCSG ▷ C3×C9
UCSC1 ◁ C2

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊AGL(1,ℤ/9))⋊C4, #6192⋊C3, #6195⋊C3, …
maximal normal subgroup of(C3⋊AGL(1,ℤ/9))⋊C4, #6192⋊C3, #6195⋊C3, …
maximal quotient of#10619962, C4×C3⋊AGL(1,ℤ/9), C4⋊(C3⋊AGL(1,ℤ/9)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 146);