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C3⋊ΓL(1,𝔽64) #11765744

order1134
isoclinismisoclinic class of C3⋊ΓL(1,𝔽64)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊ΓL(1,𝔽64)

Splittings

Fitting, derivedC3×C7×C9 ⋊ C2×C3
HallC3×#4428 ⋊ C2
SylowC7 ⋊ C3⋊AGL(1,ℤ/9)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C7×C9C2×C3
FrattiniC3C3⋊D3×ΓL(1,𝔽8)
socleC32×C7C3×D3
LCSG ▷ C3×C7×C9
UCSC1

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 189

Polycyclic presentation:

2,3^3,7,3|d^3=f,c^a=c^2,d^a=d^2f^2,f^a=f^2,d^b=df,e^b=e^4

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72229838, ΓL(1,𝔽8)×C3⋊AGL(1,ℤ/9), #72229990, …
maximal normal subgroup ofΓL(1,𝔽8)×C3⋊AGL(1,ℤ/9), #72229990, D3×ΓL(1,𝔽64), …
maximal quotient of#72229838, #72249185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1134, 174);