You can find the construction of the triple covering for the alternating group $\mathbb{A}_6$ in Chapter 2, Section 7.3 of Wilson's book:
Wilson, Robert A. The finite simple groups. Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009. xvi+298 pp. ISBN: 978-1-84800-987-5 MR2562037 (2011e:20018)
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