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C₁₃×AGL(1,𝔽₅) #63119

order	260
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₃×C₄×AGL(1,𝔽₅) (order 240) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₄×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

C₁₃×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₅×C₁₃ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₁₃
Frattini	C₁	G
Fitting, socle	C₅×C₁₃	C₄
prime radical	D₅×C₁₃	C₂
cosocle	D₅	C₂×C₁₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 65

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18);(1,2,3,5,4)

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₁₃, C₂×C₁₃×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₃⋊C₃, …
maximal normal subgroup of	AGL(1,𝔽₅)×D₁₃, C₂×C₁₃×AGL(1,𝔽₅), AGL(1,𝔽₅)×C₁₃⋊C₃, …
maximal quotient of	C₁₃×#170, C₂×C₁₃×AGL(1,𝔽₅), C₃×C₁₃×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(260, 7);