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D7×AGL(1,ℤ/9) #10625460

order756
isoclinismisoclinic class of D7×AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productD7×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×D7)
unambiguous semidirect product(D7×3−1+2)⋊C2
unambiguous semidirect product(C7×AGL(1,ℤ/9))⋊C2
unambiguous semidirect product(3−1+2⋊D7)⋊C2

Splittings

Fitting, HallC7×3−1+2 ⋊ C22
derivedC7×C9 ⋊ C22×C3
Sylow3−1+2 ⋊ C2×D7
SylowC7 ⋊ C2×AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC7×C9C22×C3
FrattiniC3C3×D3×D7
FittingC7×3−1+2C22
socleC3×C7C2×C3×D3
LCSG ▷ C7×C9
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 63

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D7×AGL(1,ℤ/9), AGL(1,𝔽7)×AGL(1,ℤ/9), AGL(1,ℤ/9)×C7⋊D7, …
maximal normal subgroup ofC2×D7×AGL(1,ℤ/9), AGL(1,𝔽7)×AGL(1,ℤ/9), C7×D7×AGL(1,ℤ/9), …
maximal quotient of#13094347, Dic7×AGL(1,ℤ/9), (3−1+2⋊D7)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 91);