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Cayley diagram

C4×AGL(1,ℤ/9) #6216

order216
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C22×AGL(1,ℤ/9) (order 216) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C22
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×C4)
unambiguous semidirect product(C4×D9)⋊C3

Splittings

FittingC4×3−1+2 ⋊ C2
derivedC9 ⋊ C2×C3×C4
Sylow3−1+2 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4AGL(1,ℤ/9)
commutator, hypocenterC9C2×C3×C4
Frattini, socleC2×C3C2×C3×D3
FittingC4×3−1+2C2
prime radicalC2×AGL(1,ℤ/9)C2
cosocleC2×C9C22×C3
LCSG ▷ C9
UCSC1 ◁ C4

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×AGL(1,ℤ/9), #3663⋊C3, C2×C4×AGL(1,ℤ/9), …
maximal normal subgroup ofC8×AGL(1,ℤ/9), #3663⋊C3, C2×C4×AGL(1,ℤ/9), …
maximal quotient ofC8×AGL(1,ℤ/9), #3663⋊C3, C4×3−1+2⋊C4, …
automorphism group ofC5×D9, C5×AGL(1,ℤ/9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 53);