What is the mathematical operation corresponding to this definition of the Dirichlet Character group operation?

The Wolfram Language & System Documentation Center page for DirichletCharacter indicates Dirichlet characters modulo k form a group:

G[k_] := Table[DirichletCharacter[k, j, n], {j, EulerPhi[k]}]

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The group operation is defined in terms of the k and j indices as follows:

add[DirichletCharacter[k_, j1_, n_], 
  DirichletCharacter[k_, j2_, n_]] := 
 DirichletCharacter[k, Mod[j1 + j2, EulerPhi[k], 1], n]

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Question: What is the mathematical operation corresponding to this definition of the group operation?

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I tried the following definition which seems to produce a different result than the definition above. Whereas the definition above is based on k and j indices the definition below assumes char1 and char2 are of the form defined by char below.

addChar[char1_, char2_] := char1 char2

char[k_, j_] := Table[DirichletCharacter[k, j, n], {n, 1, k}]

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The following evaluation illustrates addChar produces a different result than add:

Block[{k = 5, j1 = 2, j2 = 3, addResult},
 addResult = 
  addOld[DirichletCharacter[k, j1, n], DirichletCharacter[k, j2, n]];
 {Table[addResult, {n, 1, k}], addChar[char[k, j1], char[k, j2]]}]

{{1, 1, 1, 1, 0}, {1, -I, I, -1, 0}}

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Here is the list of Dirichlet characters modulo $k=5$:

j$quad$Character
1$quad$1,1,1,1,0
2$quad$1,i,-i,-1,0
3$quad$1,-1,-1,1,0
4$quad$1,-i,i,-1,0