C₂×#1132 #6465

order	224
isoclinism	isoclinic class of #1132
automorphism	Aut(G) = AGL(1,𝔽₇)×#3484 (order 5376) · Inn(G) = D₂₈ · Out(G) = C₂²×C₃×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Fitting	C₂×C₇×C₈ ⋊ C₂
Sylow	C₇ ⋊ C₂×SD₁₆

68 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)(10,11,12,14,13,15,16,17);(1,2)(4,5)(6,7)(8,9)(11,14)(12,16)(15,17);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 97);