C₂×#4294 #66784

order	342
isoclinism	isoclinic class of C₁₉⋊C₃
automorphism	Aut(G) = C₃×AGL(1,𝔽₁₉) (order 1026) · Inn(G) = C₁₉⋊C₃ · Out(G) = C₂×C₃²
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

derived, Sylow	C₁₉ ⋊ C₂×C₉
Hall	C₂×C₁₉ ⋊ C₉

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 342

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 2);