Finite Group Databasebeta

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C₈×D₇⋊C₉ #11758836

order	1008
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₈×D₇⋊C₉
unambiguous semidirect product	D₇⋊(C₈×C₉)
unambiguous semidirect product	(C₈×D₇)⋊C₉

Splittings

derived, Sylow	C₇ ⋊ C₂×C₈×C₉
Hall	C₈×D₇ ⋊ C₉
Hall	C₇⋊C₉ ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₈×C₉
Frattini	C₃×C₄	C₂×AGL(1,𝔽₇)
Fitting	C₃×C₇×C₈	C₂×C₃
socle	C₂×C₃×C₇	C₂×C₃×C₄
prime radical	C₂×C₃×D₇	C₃×C₄
cosocle	C₃×C₄×C₇	C₂²×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 504

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1008, 7);