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Parabolic permutation representations of the group $$^2E_6(q^2)$$. (English. Russian original) Zbl 0985.20008
Math. Notes 67, No. 6, 758-770 (2000); translation from Mat. Zametki 67, No. 6, 899-912 (2000).
An important class of permutation representations of groups of Lie type is formed by parabolic representations, i.e., representations on the cosets of parabolic subgroups. For finite simple groups of exceptional Lie type, the faithful parabolic representations of minimal degree were studied by A. V. Vasil’ev [see Algebra Logika 35, No. 6, 663-684 (1996; Zbl 0880.20006), ibid. 36, No. 5, 518-530 (1997; Zbl 0941.20006), ibid. 37, No. 1, 17-35 (1998; Zbl 0941.20007)]. The author [Tr. Inst. Mat. Mekh., Ural Division Russian Acad. Sci. 5, 39-59 (1998), Collect. Sci. Trans. “Combinatorial and computational methods in mathematics”, Omsk 1999, 160-189 (1999), Proc. Int. Conf. “Low-dimensional topology and combinatorial group theory” (Chelyabinsk, 1999), Kiev 38-64 (2000), and Mat. Zametki 67, No. 1, 69-76 (2000; Zbl 0965.20007)], studied the primitive parabolic representations of nonminimal degree for all finite simple groups of exceptional Lie type except for the group $$^2E_6(q^2)$$.
In the present paper, the author determines the degrees, ranks, subdegrees, and double centralizers of the permutation representations of $$^2E_6(q^2)$$ on the cosets of parabolic maximal subgroups of nonminimal index. The results obtained for the representatives $$P_2$$, $$P_3$$, and $$P_4$$ of the conjugacy classes of these subgroups are presented in tables. In these tables, for the $$l$$-th suborbit, the conjugating element $$y_l$$, the subdegree $$n_l$$, and the structure of the double centralizer $$P_j\cap P_j^{y_l}$$ corresponding to this suborbit ($$j=2,3,4$$) are indicated. The conjugating elements $$y_l$$ are explicitly indicated in the text of the paper. For computations, the software “Chevie” of the computer system GAP was used.

##### MSC:
 20C33 Representations of finite groups of Lie type 20D06 Simple groups: alternating groups and groups of Lie type 20E28 Maximal subgroups 20E45 Conjugacy classes for groups
GAP; CHEVIE
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##### References:
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