C₇×#622 #92664

order	504
isoclinism	isoclinic class of #157
automorphism	Aut(G) = C₂²×C₃×AΓL(1,𝔽₉) (order 1728) · Inn(G) = #157 · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₇×C₈ · Schur multiplier (H₂) = C₃

Aliases

derived, Sylow	C₃² ⋊ C₇×C₈
Hall	C₃²×C₇ ⋊ C₈

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 90);