C₇×Dic₉ #6964

order	252
isoclinism	isoclinic class of D₉
automorphism	Aut(G) = C₂²×C₃×AGL(1,ℤ/9) (order 648) · Inn(G) = D₉ · Out(G) = C₂²×C₃²
homology	Abelianization (H₁) = C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×Dic₉
unambiguous semidirect product	C₉⋊(C₄×C₇)

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

18 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 3);