|Algebraic structure → Group theory|
- 211 · 33 · 52 · 13 = 17,971,200.
It is sometimes considered a 27th sporadic group.
History and properties
The Ree groups 2F4(22n+1) were constructed by Ree (1961), who showed that they are simple if n ≥ 1. The first member of this series 2F4(2) is not simple. It was studied by Jacques Tits (1964) who showed that it is almost simple, its derived subgroup 2F4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2F4(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 27th sporadic group.
The Tits group occurs as a maximal subgroup of the Fischer group Fi22. The groups 2F4(2) also 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.
L3(3):2 Two classes, fused by an outer automorphism. These subgroups fix points of rank 4 permutation representations.
2..5.4 Centralizer of an involution.
A6.22 (Two classes, fused by an outer automorphism)
The Tits group can be defined in terms of generators and relations by
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