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Some details about cactus group notation.
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src/sage/groups/finitely_presented_named.py

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- Alternating group, `A_n` of order `n!/2` --
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:func:`groups.presentation.Alternating <sage.groups.finitely_presented_named.AlternatingPresentation>`
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- Cactus group, `J_n` --
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- the `n`-fruit Cactus group, a standard notation for which is `J_n` --
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:func:`groups.presentation.Cactus <sage.groups.finitely_presented_named.CactusPresentation>`
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- Cyclic group, `C_n` of order `n` --

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