Non abelian exterior square of a group. The non-abelian exterior square $G\wedge G$ of a group $G$ is defined as a group generated by the words $g\wedge h$, $g,h \in G$ related to the conditions 


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*$g\wedge g=1$ 

*$(g\wedge h)(h\wedge g)=1$

*$g\wedge g'h=(g\wedge g').(^{g'}g\wedge ^{g'}h)$

*$gg'\wedge h=(^gg'\wedge ^gh).(g\wedge h)$
where $^gh=ghg^{-1}$. The non abelian tensor square $G \otimes G$ of a group $G$ is determined by many authors For the groups of order $p^2q$ it is computed. KIndly give me some idea or refrence, so I can compute it for some groups.
 A: The standard notation for the nonabelian exterior square is $G\wedge G$ (just like the notation for the nonabelian tensor square is $G\otimes G$). The notation $G*G$ is commonly understood to represent the free product of $G$ with itself, and I strongly encourage you to drop it and adopt the standard one.
Some references:


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*Blyth, R. D.; Fumagalli, F.; Morigi, M.
A survey of recent progress on non-abelian tensor squares of groups. Ischia group theory 2010, 26–38, World Sci. Publ., Hackensack, NJ, 2012. MR3184981

*Russell D. Blyth and Robert Fitzgerald Morse. Computing the nonabelian tensor squares of polycyclic groups. J. Algebra 321 (2009), no. 8, 2139–2148. MR 2501513

*Blyth, Russell D.; Fumagalli, Francesco; Morigi, Marta.
Some structural results on the non-abelian tensor square of groups.
J. Group Theory 13 (2010), no. 1, 83–94. MR2604847 

*R. Brown, D. L. Johnson, and E. F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), no. 1, 177–202. MR0913203
See also this previous answer.
