# Minimal degree faithful permutation representations of some finite simple groups

The symmetric group $$S_n$$ - the group of permutations of an $$n$$-set - plays a very important role in Group Theory. The paramount importance of this group resides in the following fact: given any finite group $$G$$, there is a value of n such that $$S_n$$ possesses a subgroup that is structurally identical with $$G$$. The above statement is a fact.

Question: What is the minimum value of n such that $$S_n$$ possesses a subgroup that is structurally identical with the monster group $$M$$ (respectively other groups $$Fi24,B,Co1, Suz$$)?

Edit: From the comments. The question is as follows:

Question: Are the minimal degree permutation representations of the sporadic simple groups all known? I am particularly interested in the cases $$M, Fi24, B, Co1$$ and $$Suz$$.

• I'm not sure if anybody has studied these particular questions in detail. The groups you're interested in are definitely famous enough that people might have thought about this, but in general the question "what is the minimal $n$ so that $G$ embeds into $S_n$?" is extremely hard, so I wouldn't be surprised to learn there's little to nothing known about your question. Oct 29, 2021 at 5:49
• This 1998 paper states in the abstract that the minimal permutation degree is known for all finite simple groups modulo the classification. So the references therein should answer all your questions. Oct 29, 2021 at 6:01
• @DietrichBurde I think that's a little unreasonable in this case, because they are all instances of the single question "what are the minimal degree permutation representations of the sporadic finite simple groups?". I don't really understand why this question is getting so many close votes. Oct 29, 2021 at 8:05
• To the voters to close: in what respect is this question unfocused? It could be reworded "Are the minimal degree permutation representations of the sporadic simple groups all known? I am particularly interested in the cases M, Fi24, B, Co1 and Suz." Oct 29, 2021 at 8:29
• Please use more descriptive titles. Oct 29, 2021 at 9:44

It is more standard to write "isomorphic to" rather than "structurally identical with".

M: 97239461142009186000,

Fi24: 306936,

B: 13571955000,

Co1: 98280,

Suz: 1782.

The ATLAS of Finite Groups is a good source for such information, or its online version

If you looked at their Wikipedia pages you would have found this data for many of these groups.

Incidentally, the Baby Monster B was first proved to exist by Charles Sims, who constructed this representation of degree greater than $$13 \times 10^9$$ on a computer. There is a now an independent computer-free proof using the construction of M as an automorphism group of the Griess algebra.