Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties. In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is well publicised, but not the exponent, as far as I can tell.) Thanks!
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2 Answers
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I couldn't find an online table of exponents for sporadic groups, so I used GAP to produce one:
$$ \begin{align*} \mathbf{Group}&&\mathbf{Exponent}&&\mathbf{Factorization}\\ M_{11}&&1320&&2^3\cdot3\cdot5\cdot11\\ M_{12}&&1320&&2^3\cdot3\cdot5\cdot11\\ J_1&&43890&&2\cdot3\cdot5\cdot7\cdot11\cdot19\\ M_{22}&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_2&&840&&2^3\cdot3\cdot5\cdot7\\ M_{23}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ HS&&9240&&2^3\cdot3\cdot5\cdot7\cdot11\\ J_3&&116280&&2^3\cdot3^2\cdot5\cdot17\cdot19\\ M_{24}&&212520&&2^3\cdot3\cdot5\cdot7\cdot11\cdot23\\ McL&&27720&&2^3\cdot3^2\cdot5\cdot7\cdot11\\ He&&14280&&2^3\cdot3\cdot5\cdot7\cdot17\\ Ru&&633360&&2^4\cdot3\cdot5\cdot7\cdot13\cdot29\\ Suz&&360360&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ O'N&&10884720&&2^4\cdot3\cdot5\cdot7\cdot11\cdot19\cdot31\\ Co_3&&637560&&2^3\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Co_2&&1275120&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot23\\ Fi_{22}&&720720&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\\ HN&&2633400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot19\\ Ly&&10651271400&&2^3\cdot3^2\cdot5^2\cdot7\cdot11\cdot31\cdot37\cdot67\\ Th&&57886920&&2^3\cdot3^3\cdot5\cdot7\cdot13\cdot19\cdot31\\ Fi_{23}&&845404560&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\\ Co_1&&16576560&&2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot23\\ J_4&&607938537360&&2^4\cdot3\cdot5\cdot7\cdot11\cdot23\cdot29\cdot31\cdot37\cdot43\\ F_{3+}&&24516732240&&2^4\cdot3^3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot23\cdot29\\ B&&234033344344800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot31\cdot47\\ M&&1165654792878376600800&&2^5\cdot3^3\cdot5^2\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\\ &&&&\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71\\ \end{align*} $$
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1$\begingroup$ It would be more intuitive if you also paste the prime decomposition of these numbers. $\endgroup$– YCorCommented Dec 10, 2023 at 1:10
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1$\begingroup$ @JeremyRickard Thanks for adding the factoring and the nice layout. It's much better than what I posted. You should have posted it as your own answer and gotten the points. $\endgroup$– GhosterCommented Dec 10, 2023 at 20:04
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1$\begingroup$ It wasn’t me, @StevenStadnicki deserves the credit. All I did was correct the table header from “Order” to “Exponent”. $\endgroup$ Commented Dec 10, 2023 at 21:52
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1$\begingroup$ @StevenStadnicki Sorry for not paying more attention to the edit history. Thank you for your improvements! Your edit comment says that I had a typo. What was it? $\endgroup$– GhosterCommented Dec 11, 2023 at 0:42
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$\begingroup$ You didn't have a typo — my first edit did (an incorrect factorization), and I made a second edit to fix my mistake but they may have been folded into one. And you're very welcome! $\endgroup$ Commented Dec 11, 2023 at 18:39
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From the comments, This information can be calculated easily from the printed character tables in the ATLAS of Finite Groups (which include orders of elements in conjugacy classes) or, perhaps more conveniently, using the same information online via GAP or Magma. From there you can just load the character table from the library and calculate the lcm of the orders of the elements. I had a quick look at the character table of the Monster in the Atlas, and its exponent appears to be 32.27.25.7.11.13.17.19.23.29.31.47.59.71.41 - Derek Holt
Or, using the version of the ATLAS tables in Gap's character table library, Exponent(CharacterTable("F1")); (returns 1165654792878376600800) for the exponent of the Monster. – Gro-Tsen
Exponent(CharacterTable("F1"));(returns1165654792878376600800) for the exponent of the Monster. $\endgroup$