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C₂×#704 #4136

order	160
isoclinism	isoclinic class of #700
automorphism	Aut(G) = AGL(1,𝔽₅)×B(4,𝔽₂) (order 1280) · Inn(G) = C₂×AGL(1,𝔽₅) · Out(G) = #120
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₅ ⋊ C₂×M₁₆

Subgroups

	subgroup	G/H
center	C₂²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₄
Frattini	C₂	C₂²×AGL(1,𝔽₅)
Fitting, prime radical	C₂³×C₅	C₄
socle	C₂²×C₅	C₂×C₄
cosocle	Dic₅	C₂³
hypercenter	C₂³	AGL(1,𝔽₅)
hypocenter	C₅	C₂×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#64867, #64919, #65691, …
maximal normal subgroup of	#64867, #64919, #65691, …
maximal quotient of	C₂×#4002, C₄×#704, C₂×#4004, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 211);