D₄×C₇⋊C₉ #92594

order	504
isoclinism	isoclinic class of D₄×ΓL(1,𝔽₈)
automorphism	Aut(G) = C₃×D₄×AGL(1,𝔽₇) (order 1008) · Inn(G) = C₂²×ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₇⋊C₉
unambiguous semidirect product	C₄⋊(C₂×C₇⋊C₉)
unambiguous semidirect product	(C₇×D₄)⋊C₉

Hall	C₇×D₄ ⋊ C₉
Sylow	C₇ ⋊ D₄×C₉

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 252

A minimal faithful permutation representation — 2 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 20);