A sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. All the other finite simple groups form 18 infinite families numbered by q - power of prime number and n - natural number. Sporadic groups attach attention due to their sporadic/exceptional nature - similar to exceptional Lie groups. The first sporadic groups were found by Mathieu in 1860s. The last sporadic group J4 was discovered in 1975 by Janko.

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### Fusing the $\mathrm{PGL}(2,11)$ conjugacy classes of $\mathrm{Aut}(M_{12})$

Is there an embedding of $\mathrm{Aut}(M_{12})$ into the automorphism group of some larger sporadic group that fuses its two conjugacy classes of $\mathrm{PGL}(2,11)$ subgroups?
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### Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$

How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?
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### Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?

The background: We recall/define the following: $\Omega_n=\{1,\dots,n\}$. $M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups ...
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### What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt}$?

Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt}$? Indeed, they must exist. What are the simplest such polytopes ...
275 views

### A lattice with Monster group symmetries

The book Mathematical Evolutions contains the following excerpt: A last, famous, example is the following. It is known that in the space of one hundred and ninety six thousand eight hundred and ...
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### What is the geometric shape of the Monster sporadic group?

Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree." My question is, What do we know (or ...
1 vote
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### Where can I find a table of the exponents of the sporadic groups?

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 ...
1 vote
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Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups. By GAP, the set of all the ...
110 views

### Are all exceptional Schur covers sub-sporadic?

Famously, all but finitely many finite simple groups are (cyclic or alternating or) of Lie type. The groups of Lie type have central extensions coming from the simply connected covers of the ...
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### Fixed points of the automorphisms of sporadic groups

Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...
184 views

### Normalizers of abelian Sylows in simple groups

Suppose $G$ is a (nonabelian) finite simple group and $p$ is a prime such that the $p$-Sylow in $G$ is abelian. What can be said about its normalizer? I'm particularly interested in lower bounds on ...
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### Schreier conjecture -- without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...
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The dimensions of some representations of the Janko group J1 coinside with dimensions of smallest representations of the Lie algebra of type E7 (56, 133). It seems to be natural that there is a ...
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### Algebra for the Baby

I am reading the following article. Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012.. Author works with 4370-... 856 views

### Construction of representations of the Mathieu groups?

The Mathieu groups are beautiful simple finite groups. (They were the first sporadic groups to be discovered in 1861-1870). They are related with many other miraculous constructions in mathematics: ...
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### Atlas of finite groups, Character table of automorphism group of sporadic group

I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group. I am reading from Inverse Galois Theory by G. Malle Let me start with $G=M_{12}$ This(image ...
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### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...
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### Why do sporadic simple groups have so few conjugacy classes?

In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ...
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### How many sporadic simple groups are there, really?

I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...
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### (weak?) BN-Pair / Tits System for Sporadic Groups

The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and ...
351 views

### Geometric interpretation of $2A$ conjugacy class in Conway group $Co_1$

I am struggling with following problem. Consider $2A$ class in $Co_1$ having $819*759*75$ elements. Each element $a$ from $2A$ have two representatives in $Co_0$. Element $a$ corresponds to $E_8$ ... 