C₂×C₇×Q₈ #1166

order	112
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×#1020⋊C₂ (order 1152) · Inn(G) = C₂² · Out(G) = C₂²×C₃×S₄
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

C₂×C₇×Q₈

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ G

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2)(11,12,13,14,15,16,17);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 39);