order	164
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₅×C₈ (order 80) · Inn(G) = C₁ · Out(G) = C₂×C₅×C₈
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₄₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₆₄
abelian	C₄×C₄₁

Subgroups

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 164

A minimal faithful permutation representation — 1 generator on 45 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(164, 2);