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D₇×C₁₇⋊AGL(1,𝔽₇) #23756192

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

Aliases

direct product	D₇×C₁₇⋊AGL(1,𝔽₇)
unambiguous semidirect product	(D₇×ΓL(1,𝔽₈))⋊D₁₇

Splittings

Fitting, socle, derived, Hall	C₇²×C₁₇ ⋊ C₂²×C₃
Hall	D₇×D₁₁₉ ⋊ C₃
Hall	C₇×C₁₇×ΓL(1,𝔽₈) ⋊ C₂²
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×D₁₇
Sylow	C₇² ⋊ C₂×C₃×D₁₇
Sylow	C₁₇ ⋊ D₇×AGL(1,𝔽₇)

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

235 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×C₁₇⋊AGL(1,𝔽₇), C₃×D₇×C₁₇⋊AGL(1,𝔽₇), D₇×C₃⋊(C₁₇⋊AGL(1,𝔽₇)), …
maximal normal subgroup of	AGL(1,𝔽₇)×C₁₇⋊AGL(1,𝔽₇), C₃×D₇×C₁₇⋊AGL(1,𝔽₇)
maximal quotient of	D₇×C₁₇⋊(D₇⋊C₉), C₃×D₇×C₁₇⋊AGL(1,𝔽₇), D₇×C₃⋊(C₁₇⋊AGL(1,𝔽₇)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 95);