Finding 59:29 in the Monster

Abstract

The classification of the maximal subgroups of the Monster group has been completed recently, and explicit generators for each such subgroup (up to conjugacy) have been made available for the software mmgroup, with the exception of the maximal subgroup 59:29, see Dietrich et al. (Adv. Math., 2025; J Algebra, 2026). We provide explicit generators for this last maximal subgroup and comment on the extensive search that led to finding them. Our method is similar to, but significantly more involved than Bray et al.'s (London Math. Soc. J. Comput. Math., 2016) approach for constructing 47:23 in the Baby Monster. Our result allows us to provide a new short proof that the Monster does not have a subgroup PSL2(59), correcting a result of Holmes and Wilson (J. London Math. Soc., 2004).

Disclosure

“ry job took about 2.5 hours using one CPU slot. In total, approximately 43 hours elapsed from launch- ing the full first-stage search to obtaining the generators. Remark 18. Our computational search was carried out largely autonomously by AI systems: GPT-5.6 Sol Pro produced the initial design blueprint, which OpenAI Codex then audited against mmgroup; this identified and corrected substantive problems, and eventually led to the final computational pipeline. Codex wrote the required C”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-23T19:15:14.986176Z

Structural counts