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Cayley diagram

C7×AGL(1,ℤ/9) #67382

order378
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×C3×AGL(1,ℤ/9) (order 324) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×C7)
unambiguous semidirect product3−1+2⋊(C2×C7)

Splittings

Fitting, HallC7×3−1+2 ⋊ C2
derivedC9 ⋊ C2×C3×C7
Sylow3−1+2 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9C2×C3×C7
FrattiniC3C3×D3×C7
FittingC7×3−1+2C2
socleC3×C7C3×D3
LCSG ▷ C9
UCSC1 ◁ C7

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,3,4,5,6,7)(9,13,16,11,12,15)(10,14);(8,16,11,14,12,13,10,9,15)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,ℤ/9), C2×C7×AGL(1,ℤ/9), C7×#4183, …
maximal normal subgroup ofD7×AGL(1,ℤ/9), C2×C7×AGL(1,ℤ/9), ΓL(1,𝔽8)×AGL(1,ℤ/9), …
maximal quotient ofC7×3−1+2⋊C4, C2×C7×AGL(1,ℤ/9), C7×#4169, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 35);