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C₅×#700 #87842

order	400
isoclinism	isoclinic class of #700
automorphism	Aut(G) = C₄×D₄×AGL(1,𝔽₅) (order 640) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₅² ⋊ M₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄×C₅
Frattini	C₂	C₂×C₅×AGL(1,𝔽₅)
Fitting	C₄×C₅²	C₄
socle	C₂×C₅²	C₂×C₄
prime radical	C₂×C₅×D₅	C₄
cosocle	Dic₅	C₂²×C₅
hypercenter	C₄×C₅	AGL(1,𝔽₅)
hypocenter	C₅	C₅×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅

43 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 80

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11716738, #11716740, #11716744, …
maximal normal subgroup of	#11716738, #11716740, #11716744, …
maximal quotient of	C₅×C₄⋊#170, C₅×#4002, C₅×#4003, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 136);