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\(\text{PSp}_4(3)\) as a symmetric (36, 15, 6)-design. (English) Zbl 0935.05007

Let \(G\) be a group isomorphic to \(\text{PSp}_4(3) \cong U_4(2)\) and let \(\mathcal P\) be the conjugacy class of subgroups of \(G\) isomorphic to \(S_6 \cong \text{Sp}_4(2)'\). Then \(S \in {\mathcal P}\) has three orbits \(\{S\}, B\) and \(C\) on \(\mathcal P\) of length \(1,15\) and \(20\), respectively. It is shown that \(({\mathcal P}, {\mathcal B}, I)\) is a symmetric \(2\)-\((36,15,6)\) design with set of blocks \({\mathcal B} = \{B^g \mid g \in G\}\) where a point \(R\) is incident with a block \(B^g\) if \(R\) is contained in \(B^g\). The automorphism group \(\operatorname{Aut}(G)\) acts on this design flag-transitively.

MSC:

05B05 Combinatorial aspects of block designs
05E20 Group actions on designs, etc. (MSC2000)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
51E05 General block designs in finite geometry
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References:

[1] J.H. Conway - R.T. Curtis - S.P. Norton - R.A. Parker - R.A. Wilson , Atlas of Finite Groups , Oxford ( 1985 ). MR 827219 | Zbl 0568.20001 · Zbl 0568.20001
[2] Z. Janko , A Characterization of the finite simple group PSp4 (3 ), Canadian J. Math. , 19 ( 1967 ), pp. 872 - 894 . MR 214672 | Zbl 0178.02202 · Zbl 0178.02202 · doi:10.4153/CJM-1967-082-9
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