The Miller indices are a notation system used in crystallography to describe the orientation of crystal planes within a crystal lattice. These indices indicate the intercepts made by a plane with the crystallographic axes. In this case, the intercepts are made on the axes 2a, 3b, and 2c.

To determine the Miller indices for the intercepts made by a plane on crystallographic axes, we follow a set of rules:

1. Determine the intercepts: Identify the points where the plane intersects the crystallographic axes. In this case, the intercepts are made on axes 2a, 3b, and 2c.

2. Convert to fractional intercepts: Divide the intercepts by the corresponding lattice constants. For example, if the intercept on axis a is 2a, and the lattice constant along a is a, then the fractional intercept is 2.

3. Take reciprocals: Take the reciprocals of the fractional intercepts to obtain the Miller indices. For example, if the fractional intercept along a is 2, then the Miller index for axis a is 1/2.

4. Simplify the indices: If possible, simplify the Miller indices by multiplying them by a common factor to obtain the smallest integer values. For example, if the Miller index for axis a is 1/2 and the Miller index for axis c is 1/2, then the simplified Miller indices are 1 and 1.

Based on these rules, the Miller indices for the intercepts made by the plane on crystallographic axes 2a, 3b, and 2c would be (1/2, 1, 1/2) or (1, 2, 1) after simplification.

In summary, the Miller indices for the intercepts made by a plane on crystallographic axes 2a, 3b, and 2c are (1, 2, 1) after simplification. These indices provide a concise way to describe the orientation of crystal planes within a lattice, allowing crystallographers to study and understand the structure and properties of crystals.