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C3⋊(C3⋊(C3⋊AGL(1,𝔽7))) #11765813

order1134
isoclinismisoclinic class of C3⋊(C3⋊(C3⋊AGL(1,𝔽7)))
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C33

Aliases

unambiguous semidirect productC3⋊(C3⋊(C3⋊AGL(1,𝔽7)))
unambiguous semidirect productΓL(1,𝔽8)⋊(C3⋊(C3⋊D3))

Splittings

Fitting, socle, derivedC33×C7 ⋊ C2×C3
HallC33×ΓL(1,𝔽8) ⋊ C2
SylowC7 ⋊ C3×C3⋊(C3⋊D3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC33×C7C2×C3
LCSG ▷ C33×C7
UCSC1

380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 189

Polycyclic presentation:

2,3^4,7|c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^6,f^b=f^4

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊D3×C3⋊AGL(1,𝔽7), D3×C3⋊(C3⋊AGL(1,𝔽7)), AGL(1,𝔽7)×C3⋊(C3⋊D3), …
maximal normal subgroup ofC3⋊D3×C3⋊AGL(1,𝔽7), D3×C3⋊(C3⋊AGL(1,𝔽7)), AGL(1,𝔽7)×C3⋊(C3⋊D3), …
maximal quotient of#72230463, #72254861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1134, 243);