²F₄(q) - Knowlpedia

For $q=2^{2m+1}$ , where $m\geq1$ is an integer, ${}^2F_4(q)$ is the Steinberg fixed-point group

{}^2F_4(q)=\mathbf G_{F_4,\mathrm{sc}}^F,\qquad F=\tau F_{2^m},\qquad \tau^2=F_2.

Here $\mathbf G_{F_4,\mathrm{sc}}$ is the split simply connected algebraic group of type $F_4$ over $\overline{\mathbb F}_2$ , and $\tau$ is its exceptional endomorphism interchanging the long and short root data. It commutes with Frobenius. Thus $F^2=F_q$ ; the group consists of all $g$ with $F(g)=g$ , under inherited multiplication. Its center is trivial.

Order and simplicity

Its order is

|{}^2F_4(q)|=q^{12}(q^6+1)(q^4-1)(q^3+1)(q-1).

It is a nonabelian simple group for every admitted parameter.

Parameter convention

The absolute root-system rank is $4$ , and the exponent of $2$ in $q$ must be odd. The notation does not define this family over arbitrary prime powers.

The prime-field boundary

At $m=0$ , the group ${}^2F_4(2)$ has order 35,942,400. Its commutator subgroup has index two and is the Tits group, of order 17,971,200. The Tits group is displayed separately, and is not a 27th sporadic group.

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.
B. Akbari, ODs-characterization of some low-dimensional finite classical groups, Table 2, p. 85: order column only; its additional Restrictions concern a separate invariant.