order	174
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₄×D₃×C₇ (order 168) · Inn(G) = D₃ · Out(G) = C₄×C₇
holomorph	G ⋊ Aut(G) = D₃²×AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = C₂×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₂₉
unambiguous semidirect product	C₃⋊(C₂×C₂₉)

Splittings

Fitting, socle, Hall	C₃×C₂₉ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₂₉

	subgroup	G/H
center, hypercenter	C₂₉	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₂₉
Frattini	C₁	G
Fitting, socle	C₃×C₂₉	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 87

A minimal faithful permutation representation — 2 generators on 32 points:

(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);(1,2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(174, 1);