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C₃×C₇×AGL(1,𝔽₅) #88295

order	420
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) = C₃×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₇×AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₇ ⋊ C₄
derived, Sylow	C₅ ⋊ C₃×C₄×C₇
Hall	C₅×C₇ ⋊ C₃×C₄
Hall	C₃×C₅ ⋊ C₄×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₇	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃×C₄×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇	C₄
prime radical	C₃×C₇×D₅	C₂
cosocle	D₅	C₂×C₃×C₇
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 105

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₇×AGL(1,𝔽₅), D₃×C₇×AGL(1,𝔽₅), AGL(1,𝔽₅)×D₂₁, …
maximal normal subgroup of	C₃×D₇×AGL(1,𝔽₅), D₃×C₇×AGL(1,𝔽₅), AGL(1,𝔽₅)×D₂₁, …
maximal quotient of	C₃×C₇×#170, C₂×C₃×C₇×AGL(1,𝔽₅), C₇×C₉×AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 20);