C₂×#1144 #6506

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) = C₂²

Aliases

C₇ ⋊ C₂×Q₁₆

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 25 points:

(3,4,6,8,9,7,5)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(4,5)(6,7)(8,9)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 138);