C₂×C₇⋊S₄ #66746

order	336
isoclinism	isoclinic class of C₇⋊S₄
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×C₇⋊S₄
unambiguous semidirect product	(C₂×C₇)⋊S₄
unambiguous semidirect product	(C₂×A₄)⋊D₇

	subgroup	G/H
center, hypercenter	C₂	C₇⋊S₄
commutator, cosocle, hypocenter	C₇×A₄	C₂²
Frattini	C₁	G
Fitting, socle	C₂³×C₇	D₃
LCS	G ▷ C₇×A₄
UCS	C₁ ◁ C₂

66 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

(3,4,6,5)(8,9)(10,11)(12,13);(1,2)(4,5,6)(7,8,10,12,13,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 215);