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C₃×C₅×AGL(1,𝔽₅) #64485

order	300
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×C₄×AGL(1,𝔽₅) (order 160) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₅×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃×C₅² ⋊ C₄
derived	C₅ ⋊ C₃×C₄×C₅
Sylow	C₅² ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅²	C₄
prime radical	C₃×C₅×D₅	C₂
cosocle	D₅	C₂×C₃×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₅

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₅×AGL(1,𝔽₅), C₅×D₃×AGL(1,𝔽₅), AGL(1,𝔽₅)×D₁₅, …
maximal normal subgroup of	C₃×D₅×AGL(1,𝔽₅), C₅×D₃×AGL(1,𝔽₅), AGL(1,𝔽₅)×D₁₅, …
maximal quotient of	C₃×C₅×#170, C₂×C₃×C₅×AGL(1,𝔽₅), C₅×C₉×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 28);