Higman-Sims group

In mathematics, the Higman-Sims group is a finite sporadic simple group of order 44352000. It can be characterized as the simple subgroup of index two in the group of automorphisms of the Higman-Sims graph. The Higman-Sims graph has 100 nodes, so the Higman-Sims group, or HS, has a permutation representation of degree 100. The Conway groups Co₂ and Co_{3 also contain HS.}

HS can also be defined in terms of generators a and b and the following relations:

a² = b⁵ = (ab)¹¹ = (ab²)¹⁰ = [a,b]⁵ = [a,b²]⁶ = [a,bab]³ =

ababab²ab ^{- 1}ab ^{- 2}ab ^{- 1}ab²abab(ab ^{- 2})⁴ =

ab(ab²(ab ^{- 2})²)²ab²abab²(ab ^{- 1}ab²)² =

abab(ab²)²ab(ab ^{- 1})²ab(ab²)²ababab ^{- 2}ab ^{- 1}ab ^{- 2} = 1.

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