C₃×#1144 #66603

order	336
isoclinism	isoclinic class of #1141
automorphism	Aut(G) = C₂²×D₄×AGL(1,𝔽₇) (order 1344) · Inn(G) = C₂²⋊D₇ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Hall	C₃×C₇ ⋊ Q₁₆
Sylow	C₇ ⋊ C₃×Q₁₆

36 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,4,6,7,5,3)(8,18,12,10)(9,22,15,13)(11,17,21,20)(14,16,19,23)(24,25,26);(2,3)(4,5)(6,7)(8,15,12,9)(10,23,18,16)(11,14,21,19)(13,17,22,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 72);