D₃×C₃⋊(D₁₉⋊C₃) #72228802

order	2052
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₁₉	C₂²×C₃
LCS	G ▷ C₃²×C₁₉
UCS	C₁

108 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 108 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 114

A minimal faithful permutation representation — 2 generators on 25 points:

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

maximal subgroup of	D₃×C₃⋊AGL(1,𝔽₁₉), D₃²×D₁₉⋊C₃
maximal normal subgroup of	D₃×C₃⋊AGL(1,𝔽₁₉), D₃²×D₁₉⋊C₃

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,6,7,3,4,8)(5,13,14,9,10,18)(11,19,17,16,15,12)(21,22)(23,24), (1,2,5,10,15,18,17,7,6,11,16,4,8,12,14,19,13,9,3)(20,21,22)(24,25));