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D₇×ΓL(1,𝔽₈)×D₁₇ #23756199

order	9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

D₇×ΓL(1,𝔽₈)×D₁₇

Splittings

Fitting, socle, derived, Hall	C₇²×C₁₇ ⋊ C₂²×C₃
Hall	C₇×D₇×D₁₇ ⋊ C₃
Hall	C₇×C₁₇×ΓL(1,𝔽₈) ⋊ C₂²
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×D₁₇
Sylow	C₇² ⋊ C₂×C₃×D₁₇
Sylow	C₁₇ ⋊ C₂×D₇×ΓL(1,𝔽₈)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₇	C₂²×C₃
LCS	G ▷ C₇²×C₁₇
UCS	C₁

84 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×D₁₇×AGL(1,𝔽₇), C₃×D₇×ΓL(1,𝔽₈)×D₁₇, ΓL(1,𝔽₈)×D₁₇×D₂₁, …
maximal normal subgroup of	ΓL(1,𝔽₈)×D₁₇×AGL(1,𝔽₇), C₃×D₇×ΓL(1,𝔽₈)×D₁₇
maximal quotient of	D₇×D₁₇×C₇⋊C₉, C₃×D₇×ΓL(1,𝔽₈)×D₁₇, ΓL(1,𝔽₈)×D₁₇×D₂₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 102);