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Cayley diagram

C2×AGL(1,ℤ/9) #1099

order108
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×AGL(1,ℤ/9) (order 108) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2
holomorphG ⋊ Aut(G) = #72228965
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C2×C3)
unambiguous semidirect product3−1+2⋊C22
unambiguous semidirect product(C2×D9)⋊C3
unambiguous semidirect product(C2×3−1+2)⋊C2

Splittings

FittingC2×3−1+2 ⋊ C2
derivedC9 ⋊ C22×C3
Sylow3−1+2 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C22×C3
FrattiniC3C2×C3×D3
FittingC2×3−1+2C2
socleC2×C3C3×D3
LCSG ▷ C9
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AGL(1,ℤ/9), C4⋊AGL(1,ℤ/9), #611⋊C3, …
maximal normal subgroup ofC4×AGL(1,ℤ/9), C4⋊AGL(1,ℤ/9), #611⋊C3, …
maximal quotient of3−1+2⋊Q8, C4×AGL(1,ℤ/9), C4⋊AGL(1,ℤ/9), …
automorphism group ofDic9, C2×D9, C3×D9, …
holomorph ofC2×C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 26);