C₂×C₅×Q₁₆ #4120

order	160
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₄×#23882 (order 1024) · Inn(G) = D₄ · Out(G) = C₄×#120
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

C₂×C₅×Q₁₆

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

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(19,20,21,22,23);(3,4,5,6,7,8,9,10)(11,18,17,16,15,14,13,12);(3,11,7,15)(4,12,8,16)(5,13,9,17)(6,14,10,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 195);