Finite Group Databasebeta

← Prev (#63328) Next (#63330) →

C₂×C₇×AGL(1,𝔽₅) #63329

order	280
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂²×C₃×AGL(1,𝔽₅) (order 240) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₇×AGL(1,𝔽₅)
unambiguous semidirect product	D₅⋊(C₄×C₇)

Splittings

Fitting, socle	C₂×C₅×C₇ ⋊ C₄
derived, Sylow	C₅ ⋊ C₂×C₄×C₇
Hall	C₅×C₇ ⋊ C₂×C₄

Subgroups

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

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

Conjugacy classes & character table

70 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	C₇×D₅⋊C₈, C₇×#700, C₄×C₇×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(280, 34);