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C₂×C₇⋊AGL(1,𝔽₅) #63330

order	280
isoclinism	isoclinic class of C₇⋊AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 1680) · Inn(G) = C₇⋊AGL(1,𝔽₅) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇⋊AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊Dic₇
unambiguous semidirect product	(C₂×C₇)⋊AGL(1,𝔽₅)

Splittings

Fitting, socle	C₂×C₅×C₇ ⋊ C₄
derived, Hall	C₅×C₇ ⋊ C₂×C₄
Sylow	C₇ ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₂×Dic₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₇⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₂×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₇	C₄
prime radical	C₂×C₇×D₅	C₂
cosocle	C₇×D₅	C₂²
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 70

A minimal faithful permutation representation — 2 generators on 14 points:

(4,5,6,7)(9,10)(11,12)(13,14);(1,2)(3,4,7,5,6)(8,9,11,13,14,12,10)

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×Dic₇, (C₂×D₇)⋊AGL(1,𝔽₅), Dic₇⋊AGL(1,𝔽₅), …
maximal normal subgroup of	AGL(1,𝔽₅)×Dic₇, (C₂×D₇)⋊AGL(1,𝔽₅), Dic₇⋊AGL(1,𝔽₅), …
maximal quotient of	D₅⋊(C₇⋊C₈), #10587931, C₄×C₇⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 35);