2E6

2E6 may refer to:

2E6 (mathematics)

In mathematics, ²E₆ is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E₆, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as ²E₆(K) (thinking of ²E₆ as an algebraic group taking values in K) and some as ²E₆(L) (thinking of the group as a subgroup of E₆(L) fixed by an outer involution).

Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).

Over finite fields

The group ²E₆(q²) has order q³⁶ (q¹² − 1) (q⁹ + 1) (q⁸ − 1) (q⁶ − 1) (q⁵ + 1) (q² − 1) /(3,q + 1). This is similar to the order q³⁶ (q¹² − 1) (q⁹ − 1) (q⁸ − 1) (q⁶ − 1) (q⁵ − 1) (q² − 1) /(3,q − 1) of E₆(q).

Its Schur multiplier has order (3, q + 1) except for ²E₆(2²), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of ²E₆(2²) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.