How would you identify the period, block and group of the element with the electron configuration `[Ar]3d^7 4s^2`?

The symbol `[Ar]` in the condensed electron configuration of this element resembles the electron configuration of a ground-state argon `Ar` atom. It represents all inner shell electrons of this atom. (see this problem for details about condensed electron configurations.)

On the top of that, a ground-state atom of this element would contain valence electrons `3d^7 4s^2`. It would lie in the period right underneath the one containing argon. Argon is the last element of the third period. As a result, this element is in the fourth period of the periodic table. (And hence `"n" = 4`)

The block on the periodic table an element belongs to is dependent on the type of the occupied electron orbital of highest potential energy.

commons.wikimedia.org/wiki/File:Aufbau_Principlepng

Referring to the Aufbau Diagram above, the electron of the highest potential energy in a ground-state atom of this element lies in a `4d` orbital (the one in an orbital that is reached after all other occupied atomic orbitals.) That element, therefore, is located in the `d` block of the periodic table.

An atom of this element contains `9` valence electrons in the ground state. It would thus be in IUPAC Group `9` of the periodic table, which corresponds to old IUPAC Group `"VIII"`.

https://en.wikipedia.org/wiki/Group_(periodic_table)

References:
"Group (periodic table)", English Wikipedia, https://en.wikipedia.org/wiki/Group_(periodic_table)

"Block (periodic table)", English Wikipedia,
https://en.wikipedia.org/wiki/Block_(periodic_table)