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C₃×AGL(1,𝔽₅) #305

order	60
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅) (order 40) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂
holomorph	G ⋊ Aut(G) = D₃×AGL(1,𝔽₅)²
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

C₃×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃×C₅ ⋊ C₄
derived, Sylow	C₅ ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅	C₄
prime radical	C₃×D₅	C₂
cosocle	D₅	C₂×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 15

A minimal faithful permutation representation — 2 generators on 8 points:

(2,5,4,3)(6,7,8);(1,2,3,5,4)

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅), C₂×C₃×AGL(1,𝔽₅), C₉×AGL(1,𝔽₅), …
maximal normal subgroup of	D₃×AGL(1,𝔽₅), C₂×C₃×AGL(1,𝔽₅), C₉×AGL(1,𝔽₅), …
maximal quotient of	C₃×#170, C₂×C₃×AGL(1,𝔽₅), C₉×AGL(1,𝔽₅), …
outer automorphism group of	D₇×5₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60, 6);