# Tits group

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In mathematics, the **Tits group** ^{2}*F*_{4}(2)′ (Template:IPA-fr) is a finite simple group of order 17971200 = 2^{11} · 3^{3} · 5^{2} · 13 found by Template:Harvs.

The Ree groups ^{2}*F*_{4}(2^{2n+1}) were constructed by Template:Harvtxt, who showed that they are simple if *n* ≥ 1. The first member of this series ^{2}*F*_{4}(2) is not simple. It was studied by Template:Harvs who showed that its derived subgroup ^{2}*F*_{4}(2)′ of index 2 was a new simple group, the Tits group. The group ^{2}*F*_{4}(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. Because the Tits group is not strictly a group of Lie type, it is sometimes regarded as a sporadic group.

## Properties

The Schur multiplier of the Tits group is trivial and its outer automorphism group has order 2, with the full automorphism group being the group ^{2}*F*_{4}(2).

The group ^{2}*F*_{4}(2) occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points.

The Tits group is one of the simple N-groups, and was overlooked in John G. Thompson's first announcement of the classification of simple *N*-groups, as it had not been discovered at the time. It is also one of the thin finite groups.

The Tits group was characterized in various ways by Template:Harvs and Template:Harvtxt.

## Maximal subgroups

Template:Harvtxt and Template:Harvtxt independently found the 8 classes of maximal subgroups of the Tits group as follows:

L_{3}(3):2 Two classes, fused by an outer automorphism. These subgroup fix points of rank 4 permutation representations.

2.[2^{8}].5.4 Centralizer of an involution.

L_{2}(25)

2^{2}.[2^{8}].S_{3}

A_{6}.2^{2} (Two classes, fused by an outer automorphism)

5^{2}:4A_{4}

## Presentation

The Tits group can be defined in terms of generators and relations by

where [*a*, *b*] is the commutator. It has an outer automorphism obtained by sending (*a*, *b*) to (*a*, *bbabababababbababababa*).

## References

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