C₂×C₃×C₁₉⋊C₃ #66794

order	342
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₂×C₃×C₁₉⋊C₃
unambiguous semidirect product	(C₂×C₁₉)⋊C₃²
unambiguous semidirect product	(C₂×C₃×C₁₉)⋊C₃

	subgroup	G/H
center, hypercenter	C₂×C₃	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₉	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₃

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 12);