C₂×C₄×C₇⋊C₉ #92593

order	504
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×D₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = ΓL(1,𝔽₈) · Out(G) = C₂×C₃×D₄
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₄×C₇⋊C₉
unambiguous semidirect product	(C₂×C₄×C₇)⋊C₉

derived, Sylow	C₇ ⋊ C₂×C₄×C₉
Hall	C₂×C₄×C₇ ⋊ C₉

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 504

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 19);