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C₂×C₅×D₇⋊C₉ #11929464

order	1260
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅×D₇⋊C₉
unambiguous semidirect product	D₇⋊(C₂×C₅×C₉)
unambiguous semidirect product	(C₂×D₇)⋊(C₅×C₉)
unambiguous semidirect product	(C₇⋊C₉)⋊(C₂²×C₅)
unambiguous semidirect product	(C₂×C₇⋊C₉)⋊(C₂×C₅)
unambiguous semidirect product	(C₂×C₅×D₇)⋊C₉

Splittings

derived, Sylow	C₇ ⋊ C₂²×C₅×C₉
Hall	C₅×C₇⋊C₉ ⋊ C₂²
Hall	C₂×C₅×D₇ ⋊ C₉
Hall	C₇⋊C₉ ⋊ C₂²×C₅
Hall	C₅×C₇ ⋊ C₂²×C₉
Hall	C₂×D₇ ⋊ C₅×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂²×C₅×C₉
Frattini	C₃	C₂×C₅×AGL(1,𝔽₇)
Fitting, socle	C₂×C₃×C₅×C₇	C₂×C₃
prime radical	C₂×C₃×C₅×D₇	C₃
cosocle	C₃×C₇	C₂²×C₃×C₅
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 630

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₁)×D₇⋊C₉, C₂×(C₁₁⋊C₅)⋊(D₇⋊C₉), C₂×C₁₁⋊C₅×D₇⋊C₉, …
maximal normal subgroup of	C₂×C₅×C₁₁×D₇⋊C₉, C₂×C₅×C₁₃×D₇⋊C₉, C₂×C₅×C₁₇×D₇⋊C₉, …
maximal quotient of	AGL(1,𝔽₁₁)×D₇⋊C₉, C₂×(C₁₁⋊C₅)⋊(D₇⋊C₉), C₂×C₁₁⋊C₅×D₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 16);