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Cayley diagram

C5×AGL(1,ℤ/9) #63193

order270
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C4×AGL(1,ℤ/9) (order 216) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C4
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×C5)
unambiguous semidirect product3−1+2⋊(C2×C5)
unambiguous semidirect product(C5×D9)⋊C3

Splittings

Fitting, HallC5×3−1+2 ⋊ C2
derivedC9 ⋊ C2×C3×C5
Sylow3−1+2 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C2×C3×C5
FrattiniC3C3×C5×D3
FittingC5×3−1+2C2
socleC3×C5C3×D3
LCSG ▷ C9
UCSC1 ◁ C5

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 45

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×AGL(1,ℤ/9), C2×C5×AGL(1,ℤ/9), C5×#4183, …
maximal normal subgroup ofD5×AGL(1,ℤ/9), C2×C5×AGL(1,ℤ/9), C3×C5×AGL(1,ℤ/9), …
maximal quotient ofC5×3−1+2⋊C4, C2×C5×AGL(1,ℤ/9), C5×#4169, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 11);