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C₂×#687 #4079

order	160
isoclinism	isoclinic class of #686
automorphism	Aut(G) = AGL(1,𝔽₅)×#3484 (order 2560) · Inn(G) = C₂²⋊D₅ · Out(G) = C₂×C₄×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

C₅ ⋊ C₂×SD₁₆

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂³
Frattini	C₄	C₂²×D₅
Fitting, prime radical	C₂×C₅×D₄	C₂
socle	C₂²×C₅	D₄
hypercenter	C₂×D₄	D₅
hypocenter	C₅	C₂×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

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:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#65037, #65046, #65047, …
maximal normal subgroup of	#65037, #65046, #65047, …
maximal quotient of	C₂×#3940, C₂×#3942, #65260, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 154);