C₇×Q₈ #291

order	56
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×S₄ (order 144) · Inn(G) = C₂² · Out(G) = C₂×C₃×D₃
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₇)×#5935
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

C₇×Q₈

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

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,4,6)(3,7,8,5)(9,10,11,12,13,14,15);(1,3,4,8)(2,5,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 10);