C₁₉×Dic₆ #90709

order	456
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) = C₂×D₃×D₄×C₉ (order 864) · Inn(G) = C₂×D₃ · Out(G) = C₂³×C₉
homology	Abelianization (H₁) = C₂²×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×Dic₆
unambiguous semidirect product	C₃⋊(Q₈×C₁₉)

Hall	C₃×C₁₉ ⋊ Q₈
Sylow	C₃ ⋊ Q₈×C₁₉

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 456

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(4,11,7,6)(5,10,9,8)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30);(2,3)(4,9,7,5)(6,8,11,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 29);