C₂×C₃×C₂₉ #4307

order	174
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₇ (order 56) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₇
holomorph	G ⋊ Aut(G) = C₂×D₃×AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₇₄
abelian	C₂×C₃×C₂₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 174

A minimal faithful permutation representation — 1 generator on 34 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)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(174, 4);