Tits group - Knowlpedia

The Tits group ${}^2F_4(2)'$ is the commutator subgroup of ${}^2F_4(2)$ :

{}^2F_4(2)'=\langle aba^{-1}b^{-1}:a,b\in {}^2F_4(2)\rangle.

Its multiplication is inherited from the specified finite group. This definition uses the full group at its exceptional small parameter before taking the commutator subgroup.

Order and simplicity

Its order is

|{}^2F_4(2)'|=17971200.

It is a nonabelian simple group. The exact order is 17,971,200.

Identification and index

It is a normal subgroup of index $2$ in ${}^2F_4(2)$ . This is the Tits group. Its order factors as $2^{11}3^3 5^2 13$ . It is a special group at the boundary of the exceptional Lie-type families. It is not counted among the 26 sporadic simple groups.

References

Robert A. Wilson, On the simple groups of Suzuki and Ree (24 April 2010), Introduction; §5.1 orders, §5.2 simplicity, and §5.3 the groups over the prime fields, pp. 24–30.
Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §11, especially Theorems 34–35, PDF pp. 193–196; notation for field and twisting parameters differs.
ATLAS of Finite Group Representations: Tits group, Group heading: order 17,971,200 = 2^11·3^3·5^2·13; extension 2F4(2)′.2.