C₃×C₇×Dic₆ #92707

order	504
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₂²×C₃×D₃×D₄ (order 576) · Inn(G) = C₂×D₃ · Out(G) = C₂⁴×C₃
homology	Abelianization (H₁) = C₂²×C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₇×Dic₆
unambiguous semidirect product	C₃⋊(C₃×C₇×Q₈)

Hall	C₃²×C₇ ⋊ Q₈
Sylow	C₃² ⋊ C₇×Q₈

54 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)(4,5,6)(7,8,9,10,11,12,13)(14,16,17,21)(15,18,19,20);(5,6)(14,15,17,19)(16,20,21,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 133);