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Cayley diagram

C2×C3×AGL(1,ℤ/9) #66448

order324
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×D3×AGL(1,ℤ/9) (order 648) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×D3
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

BorelB(2,ℤ/9)
direct productC2×C3×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C2×C32)
unambiguous semidirect product(C2×D9)⋊C32
unambiguous semidirect product(C2×C3×D9)⋊C3

Splittings

FittingC2×C3×3−1+2 ⋊ C2
derivedC9 ⋊ C22×C32
SylowC3×3−1+2 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C22×C32
FrattiniC3C2×C32×D3
FittingC2×C3×3−1+2C2
socleC2×C32C3×D3
LCSG ▷ C9
UCSC1 ◁ C2×C3

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 14 points:

(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

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×AGL(1,ℤ/9), #6192⋊C3, #6194⋊C3, …
maximal normal subgroup ofDic3×AGL(1,ℤ/9), #6192⋊C3, #6194⋊C3, …
maximal quotient ofC3×3−1+2⋊Q8, C3×C4×AGL(1,ℤ/9), C3×C4⋊AGL(1,ℤ/9), …
automorphism group ofC7×D9, C9×D9, Dic9⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 140);