order	190
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = C₄×AGL(1,𝔽₁₉) (order 1368) · Inn(G) = D₁₉ · Out(G) = C₄×C₉
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₉
unambiguous semidirect product	C₁₉⋊(C₂×C₅)

Splittings

Fitting, socle, Hall	C₅×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₅

	subgroup	G/H
center, hypercenter	C₅	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₅

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 95

A minimal faithful permutation representation — 2 generators on 24 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(190, 2);