A₄×C₂₉ #66829

order	348
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = C₄×C₇×S₄ (order 672) · Inn(G) = A₄ · Out(G) = C₂×C₄×C₇
homology	Abelianization (H₁) = C₃×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₂₉
unambiguous semidirect product	C₂²⋊(C₃×C₂₉)
unambiguous semidirect product	(C₂²×C₂₉)⋊C₃

Fitting, socle, Hall	C₂²×C₂₉ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₂₉

	subgroup	G/H
center, hypercenter	C₂₉	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₂₉
Frattini	C₁	G
Fitting, socle	C₂²×C₂₉	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂₉

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 116

A minimal faithful permutation representation — 2 generators on 33 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(1,4)(2,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(348, 8);