What are the powers of president and Prime minister of India

The powers of President : Appoints PM, ministers, Chairman and members of UPSC. Comptroller and auditor General, Chief Election Commissioner and other members of Election Commission, Governors, Member of Finance Commission, Ambassadors, Etc. Can summon and prorogue the sessions of the 2 houses and can dissolve Lok sabha. Appoints Finance Commission ( after 5 years ) that recommends distribution of taxes between Union and state Govts. Appoints the chief justice and the judge of the supreme court and High courts.
The presidents can promulgate 3 types of Emergencies :

1. Article 352 - National Emergency 
2. Article 356 - State Emergency ( President's rule )
3. Article 360 - Financial Emergency

He is the Supreme Commander of the Defence Force of India. President appoints chief of Army, Navy and Air Force. Declares War and concludes peace subject to the approval of the Parliament.

The powers of Prime Minister : Real executive authority. He is the ex-officio Chairman of the NITI Aayog, National Development Council, National Integration Council and Inter-State Council. The president convenes and prorogues all sessions of Parliament in consultation with him. Can recommend the dissolution of Lok Sabha before expiry. Appoints the Council of ministers. Allocation portfolios. Can ask minister to resign and can get him dismissed by President. Assists the President in appointment of all high officials. Can recommend to the president to declare emergency on ground of war. external aggression or armed rebellion. Advises President  about President's  Rule in the state or emergency due to financial instability. Leader of the house.

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