order	112
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = AGL(1,𝔽₇)×B(4,𝔽₂) (order 2688) · Inn(G) = C₂×D₇ · Out(G) = C₃×#120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×Dic₁₄
unambiguous semidirect product	(C₂×C₇)⋊Q₈

Splittings

C₇ ⋊ C₂×Q₈

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 27);