C₂×D₃×C₂₉ #66831

order	348
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₄×D₃×C₇ (order 336) · Inn(G) = D₃ · Out(G) = C₂×C₄×C₇
homology	Abelianization (H₁) = C₂²×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₂₉
unambiguous semidirect product	C₃⋊(C₂²×C₂₉)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₂₉)

	subgroup	G/H
center, hypercenter	C₂×C₂₉	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₂₉
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₂₉	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₉

20 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2)(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);(1,2)(3,4,5)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(348, 10);