Write the multiplication for the characters of S_5. That is, write every product of irreducible characters as a sum of irreducible characters (the table will be symmetric).
Since $\displaystyle S_5$ has 7 conjugacy classes, it has 7 irreducible characters.
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Size 1 10 20 30 24 15 20
Class 1, (12), (123), (1234), (12345), (12)(34), (12345)
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X1 1 1 1 1 1 1 1
X2 1 -1 1 -1 1 1 -1
X3 4 2 1 0 -1 0 -1
X4 4 -2 1 0 -1 0 1
X5 6 0 0 0 1 -2 0
X6 5 1 -1 -1 0 1 1
X7 5 -1 -1 1 0 1 -1
X1 is the character for the trivial representation of S_5.
X2 is the character for the alternating(sign) representation of S_5.
X3 is the standard representation of S_5.
X4 = X2$\displaystyle \cdot$X3. It turns out to be an irreducible character of S_5.
X5 = 1/2(X3(g)^2 - X3(g^2)),
X_6,
X_7 = X2$\displaystyle \cdot$X5.
These characters satisfy both "row orthogonality theorem" and "column orthogonal theorem" (link).
Now you need to do some calculations and fill the table.