order	114
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₉) (order 684) · Inn(G) = D₁₉ · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

C₃×D₁₉

Splittings

Fitting, socle, Hall	C₃×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₃	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 57

A minimal faithful permutation representation — 2 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(114, 4);