Finite Group Databasebeta

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C₂×C₇×D₇×C₄₇ #22604263

order	9212
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₄₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×D₇×C₄₇
unambiguous semidirect product	C₇⋊(C₂²×C₇×C₄₇)
unambiguous semidirect product	(C₂×C₇)⋊(C₂×C₇×C₄₇)

Splittings

Fitting, socle	C₂×C₇²×C₄₇ ⋊ C₂
derived	C₇ ⋊ C₂²×C₇×C₄₇
Hall	C₇²×C₄₇ ⋊ C₂²
Sylow	C₇² ⋊ C₂²×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇×C₄₇	D₇
commutator, hypocenter	C₇	C₂²×C₇×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₇	C₂
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇×C₄₇

52 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

3290 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₄₇×#64418, C₂×C₇×AGL(1,𝔽₇)×C₄₇, C₂×D₇×ΓL(1,𝔽₈)×C₄₇, …
maximal normal subgroup of	C₂×C₇×AGL(1,𝔽₇)×C₄₇, C₂×D₇×ΓL(1,𝔽₈)×C₄₇, C₂×C₄₇×#64423, …
maximal quotient of	D₃×C₇×D₇×C₄₇, C₂×C₃×C₇×D₇×C₄₇, C₂×C₇×D₂₁×C₄₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9212, 27);