C₂²×C₁₉⋊C₃ #6583

order	228
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₉) (order 2052) · Inn(G) = C₁₉⋊C₃ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₂²×C₁₉⋊C₃

Fitting, socle, Hall	C₂²×C₁₉ ⋊ C₃
derived, Sylow	C₁₉ ⋊ C₂²×C₃

	subgroup	G/H
center, hypercenter	C₂²	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²

20 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(228, 9);