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D₇×AΣL(1,𝔽₂₇) #11765654

order	1134
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

D₇×AΣL(1,𝔽₂₇)

Splittings

Fitting, Hall	C₇×AΣL(1,𝔽₂₇) ⋊ C₂
Sylow	C₇ ⋊ C₂×AΣL(1,𝔽₂₇)

Subgroups

	subgroup	G/H
center	C₃	D₇×H(3,𝔽₃)
commutator, cosocle	C₃²×C₇	C₂×C₃²
Frattini	C₃²	C₃²×D₇
Fitting	C₇×AΣL(1,𝔽₂₇)	C₂
socle	C₃×C₇	C₂×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	D₇
hypocenter	C₇	C₂×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃²×C₇ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 63

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₇×AΣL(1,𝔽₂₇), D₇×#6909, AGL(1,𝔽₇)×AΣL(1,𝔽₂₇), …
maximal normal subgroup of	C₃×D₇×AΣL(1,𝔽₂₇), D₇×#6909, AGL(1,𝔽₇)×AΣL(1,𝔽₂₇), …
maximal quotient of	C₃×D₇×AΣL(1,𝔽₂₇)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1134, 84);