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I think it is more common than not, except in very small countries.
Capital cities are often intentionally designated away from the largest city, typically the primary economic center, in order to remove or reduce the influence of that city's chiefs on the broader needs of the country as a whole.
You can see this in most of the sub-national capitals in the US states, as well.
In Australia all of the state capitals are the biggest cities of their respective states. Usually by a very wide margin.
Of Canada's 10 provinces, 5 have the largest city as the capital, and 5 don't.
I know there are some cases where the capital used to be the largest city, but then they built a new city to be the capital. Thus, Belize City used to be the capital of Belize until they built Belmopan. I'm pretty sure that Lagos used to be the capital of Nigeria before Abuja. I also think the same thing happened with Brasilia (Brazil), though I don't know what was the capital before then.
Metropolitan area, which comprises of suburbs also, will give a better idea of population of Capital City.
Although Scotland is not a country by UN definition, Glasgow is a bigger city then its capital Edinburgh.
Countries like India, Pakistan used to have their capital in biggest cities of those days- Calcutta and Karachi. Shifting to Delhi and Islamabad happened later on.
Beijing and Shanghai are about the same size, in terms of urban area or urban population.
(Although in the 1940s the size of Beijing was much smaller.)
No where but on these forums is the definition of a city debatable.
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