#4294⋊C₂ #66783

order	342
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	Aut(G) = C₃×AGL(1,𝔽₁₉) (order 1026) · Inn(G) = D₁₉⋊C₃ · Out(G) = C₃²
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₁₉⋊C₃
commutator, hypocenter	C₁₉	C₂×C₉
Fitting, socle, cosocle	C₃×C₁₉	C₂×C₃
prime radical	C₃×D₁₉	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₃

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 171

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2,3^2,19|b^3=c,d^a=d^18,d^b=d^7

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 1);