C₂×C₅×#41 #4156

order	160
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₄×#11713200 (order 3072) · Inn(G) = C₂² · Out(G) = C₄×D₄×S₄
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₅×#41

	subgroup	G/H
center	C₂×C₄×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₅
socle	C₂²×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ G

164 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 231);