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D₇×ΓL(1,𝔽₈)×C₄₁ #27130307

order	12054
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

D₇×ΓL(1,𝔽₈)×C₄₁

Splittings

Fitting, socle, Hall	C₇²×C₄₁ ⋊ C₂×C₃
derived, Sylow	C₇² ⋊ C₂×C₃×C₄₁
Hall	C₇×ΓL(1,𝔽₈)×C₄₁ ⋊ C₂
Hall	C₇×D₇×C₄₁ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₄₁
Hall	C₇×D₇ ⋊ C₃×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₇²×C₄₁	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄₁

34 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×C₄₁×AGL(1,𝔽₇), D₇×ΓL(1,𝔽₈)⋊D₄₁, D₇×ΓL(1,𝔽₈)×D₄₁, …
maximal normal subgroup of	D₇×C₄₁×AGL(1,𝔽₇), D₇×ΓL(1,𝔽₈)⋊D₄₁, D₇×ΓL(1,𝔽₈)×D₄₁, …
maximal quotient of	ΓL(1,𝔽₈)×Dic₇×C₄₁, C₂×D₇×ΓL(1,𝔽₈)×C₄₁, D₇×C₄₁×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12054, 29);