C₁₉×D₁₁ #88271

order	418
isoclinism	isoclinic class of D₁₁
automorphism	Aut(G) = C₂×C₉×AGL(1,𝔽₁₁) (order 1980) · Inn(G) = D₁₁ · Out(G) = C₂×C₅×C₉
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₁₁
unambiguous semidirect product	C₁₁⋊(C₂×C₁₉)

Fitting, socle, Hall	C₁₁×C₁₉ ⋊ C₂
derived, Sylow	C₁₁ ⋊ C₂×C₁₉

	subgroup	G/H
center, hypercenter	C₁₉	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₁₁×C₁₉	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 209

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(418, 1);