C₂×#1141 #6494

order	224
isoclinism	isoclinic class of #1141
automorphism	Aut(G) = AGL(1,𝔽₇)×#3484 (order 5376) · Inn(G) = C₂²⋊D₇ · Out(G) = C₂²×C₃×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

Fitting	C₂×C₇×D₄ ⋊ C₂
Sylow	C₇ ⋊ C₂×D₈

76 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)(11,14)(12,17)(15,16);(4,5)(6,7)(8,9)(10,11)(12,14)(13,16)(15,17);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 126);