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C₃×C₇⋊AGL(1,𝔽₅) #88296

order	420
isoclinism	isoclinic class of C₇⋊AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×AGL(1,𝔽₇) (order 1680) · Inn(G) = C₇⋊AGL(1,𝔽₅) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

C₃×C₇⋊AGL(1,𝔽₅)

Splittings

Fitting, socle, Hall	C₃×C₅×C₇ ⋊ C₄
derived, Hall	C₅×C₇ ⋊ C₃×C₄
Hall	C₃×C₇ ⋊ AGL(1,𝔽₅)
Hall	C₃×C₅ ⋊ Dic₇
Sylow	C₇ ⋊ C₃×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₃×Dic₇

Subgroups

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

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:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 21);