On odd degree parts of cohomology of sporadic simple groups whose Sylow \(p\)-subgroup is the extra-special \(p\)-group of order \(p^3\) (Q1270354)

Let \(p\) be an odd prime and \(G\) be a sporadic simple group whose Sylow \(p\)-subgroups are isomorphic to the extraspecial \(p\)-group \(E=p_+^{1+2}\) of order \(p^3\) and of exponent \(p\). The cohomology ring \(H^*(G;F_p)\) is a subring of \(H^*(E;F_p)\). \textit{M. Tezuka} and \textit{N. Yagita} [J. Algebra 183, No. 2, 483-513 (1996; Zbl 0855.20045)] computed \(H^{\text{even}}(G)_{(p)}\) for all sporadic groups \(G\) whose Sylow \(p\)-subgroups are \(E\). In this paper, the author computes \(H^*(G;F_p)\) by studying its \(H^{\text{even}}(G)_{(p)}\)-module structure.