Multi agent systems and consensus problems are theoretical aspect of Quadratic Stochastic Operators (QSO). The extreme doubly stochastic quadratic operators (EDSQOs) on two-dimensional simplex (2DS) exposes a complex problem within QSO and majorization theories in non-linear model. Previous research studies on EDSQOs fails to present full transition matrices and operators of EDSQOs on 2DS. Crucial to that is the classification of those operators within each permutation. In order to address these gaps, this research designed all the transition matrices for each EDSQO on 2DS under the sufficient conditions of majorization concept. Hence, the study defines all the EDSQOs on 2DS and investigate the sufficient conditions of Majorization concept for EDSQOs on 2DS. Matlab is utilize for analysis of evaluating the number of EDSQOs on 2DS. The result of the analysis of the transition matrices and operators indicates 222 EDSQOs on 2DS. Further analysis enables this research to classify the 222 EDSQOs into 37 groups of EDSQOs based on a permutation of each EDSQOs. This study has impact on a model for consensus problems and multi agent systems.