C₂×C₃×D₁₉ #6586

order	228
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₉) (order 1368) · Inn(G) = D₁₉ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₂×C₃×D₁₉

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₃

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 114

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);(1,2)(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(228, 12);