C₇×D₄ #290

order	56
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×D₄ (order 48) · Inn(G) = C₂² · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×#453
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₇×D₄
unambiguous semidirect product	C₄⋊(C₂×C₇)
unambiguous semidirect product	C₂²⋊(C₂×C₇)

	subgroup	G/H
center, socle	C₂×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ G

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 9);