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Never heard 'шестёрка' before; is that used a lot in speech?
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1 едини́ца
2 дво́йка
3 тро́йка
4 четвёрка
5 пятёрка
6 шестёрка
7 семёрка
8 восьмёрка
9 девя́тка
10 деся́тка

These words have several meanings.

They can be used to denote a group of n objects. E.g. Большая Семёрка ‘G7 summit’; тройка ‘troika, three-horse team’. This meaning is rather bookish.

Another meaning, used a lot in both literary and colloquial speech, is ‘a figure written out’.
E.g.
большая синяя единица — a big blue ‘1’.
Число зверя — три шестёрки — The Number of the Beast is three 6’s.

In particular, these words are used for school marks. (We have a five-mark grading system in Russia, пятёрка being the best and единица (also known as кол for its shape) the worst mark).

Playing cards (2, …, 10) are called двойка, …, десятка.
E.g. шестёрка червей ‘6 of hearts’.

Things which model names contain number 1—10 are also often called informally using these words:
OS Windows 7 — «семёрка», Windows 8 — «восьмёрка», Windows 10 — «десятка».
Car model ВАЗ-2106 — «шестёрка», ВАЗ-2107 — «семёрка», ВАЗ-2109 — «девятка» etc.