order	456
isoclinism	isoclinic class of D₄×C₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₁₉⋊C₃
unambiguous semidirect product	C₄⋊(C₂×C₁₉⋊C₃)
unambiguous semidirect product	(D₄×C₁₉)⋊C₃

Splittings

Fitting, Hall	D₄×C₁₉ ⋊ C₃
Sylow	C₁₉ ⋊ C₃×D₄

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18)(21,23);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)(20,21)(22,23)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 20);