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  "body": "\n## Description\n\nThe Suzuki groups $\\mathrm{Sz}(q)$, also written ${}^2B_2(q)$, are an infinite family of finite [[groups of Lie type|NPEBABQ49M]].\nThe parameter is $q = 2^{2n+1}$ with $n \\ge 0$.\nThe [[order|0B85S2V2DR]] of $\\mathrm{Sz}(q)$ is\n$$\\abs{\\mathrm{Sz}(q)} = q^2 (q^2 + 1)(q - 1).$$\nFor $n \\ge 1$ the group $\\mathrm{Sz}(q)$ is a non-abelian [[simple group|0YMYMNQ0QZ]]; the smallest one is $\\mathrm{Sz}(8)$, of order $29120$.\nThe group $\\mathrm{Sz}(2)$ has order $20$ and is not simple.\n\n## Remarks\n\n$\\mathrm{Sz}(q)$ is the subgroup of the [[symplectic group|X12FS2JAP0]] $\\mathrm{Sp}_4(\\F_q)$ fixed by a certain automorphism.\nThis automorphism combines the exceptional graph automorphism of $\\mathrm{Sp}_4$ in characteristic $2$ with a field automorphism, and it exists because $q$ is an odd power of $2$.\nIt acts doubly transitively on a set of $q^2 + 1$ points, an ovoid in the projective space of dimension $3$ over $\\F_q$.\nThe Suzuki groups are the only non-abelian finite simple groups whose order is not divisible by $3$.\nThey were found by Michio Suzuki.\nThey are not related to the sporadic Suzuki group $\\mathrm{Suz}$.\n\n## Explanations\n\n### L1 human\n\nMost finite simple groups have an order divisible by $3$.\nThe Suzuki groups are the exception: an infinite family of simple groups whose sizes are not multiples of $3$.\nThey live inside $4 \\times 4$ symplectic matrix groups over fields with $2^{2n+1}$ elements, and they exist because of a special symmetry that appears only in characteristic $2$.\n",
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