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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Which quasisimple groups with central quotient $G\cong\mathrm{PSL}_3(4)$ are isomorphic to subgroups of the Monster sporadic group? So far I know that $G$ itself is not and that $2\cdot G$, $2^2\cdot ... 6 votes 1 answer 263 views ### Which finite simple groups are rational-relative-real? A finite group$G$is called rational if every element$g \in G$is conjugate to all of its primitive powers$g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$. Analogously, I'll call$G$real ... 1 vote 1 answer 82 views ### On reducing degree-$12$equations with Mathieu group$M_{11}$to its degree-$11$resolvent? The general$4$-deg and some$8$-deg (such as the Schein octic) when a linear transformation is done so their$x^{n-1}$term vanishes can have a neat solution as,$$x = \sqrt{z_1}+\sqrt{z_2}+\sqrt{z_3}... 8 votes 1 answer 397 views ### Small subgroups of the monster Is every group of order at most 36 isomorphic to a subgroup of the monster group? 3 votes 0 answers 122 views ###$2^2 \cdot U_6(2)$and$2^2.2^{1+20}U_6(2)$in$\mathbb{M}$In the first diagram of this paper, there are conjugacy classes of subgroups of the Monster group which are labeled$2^2 \cdot U_6(2)$and$2^2.2^{1+20}U_6(2)$, respectively. Can subgroups in the ... 5 votes 0 answers 326 views ### Does$\mathit{Suz}$contain$M_{13}$?$\newcommand\Suz{\mathit{Suz}}$I recently noticed that the Suzuki group$\Suz$has as subgroups classes of both$L_3(3)$and$M_{12}$, both of which are also subgroups of the Mathieu groupoid$M_{13}$.... 2 votes 1 answer 154 views ### 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? -4 votes 1 answer 135 views ### 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? 2 votes 2 answers 216 views ### 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 ... 6 votes 1 answer 487 views ### 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} $? Indeed, they must exist. What are the simplest such polytopes ... 8 votes 0 answers 415 views ### A lattice with Monster group symmetries The book Mathematical Evolutions contains the following excerpt: A last, famous, example is the following. It is known that in the space of one hundred and ninety six thousand eight hundred and ... 20 votes 3 answers 3k views ### What is the geometric shape of the Monster sporadic group? Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree." My question is, What do we know (or ... 1 vote 1 answer 195 views ### Where can I find a table of the exponents of the sporadic groups? Is there a table showing Sporadic Groups and their exponents, and, perhaps, other basic properties. In particular, I'm interested in what the exponent of the Monster Group is. (Obviously the order is ... 1 vote 1 answer 407 views ### The sporadic numbers Let call$n$a sporadic number if the set of groups$G \neq A_n,S_n$having a core-free maximal subgroup of index$n$is non-empty and contains only sporadic simple groups. By GAP, the set of all the ... 3 votes 0 answers 122 views ### Are all exceptional Schur covers sub-sporadic? Famously, all but finitely many finite simple groups are (cyclic or alternating or) of Lie type. The groups of Lie type have central extensions coming from the simply connected covers of the ... 3 votes 1 answer 201 views ### Fixed points of the automorphisms of sporadic groups Sporadic groups have very few outer automorphisms (in fact,$|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ... 4 votes 0 answers 206 views ### Normalizers of abelian Sylows in simple groups Suppose$G$is a (nonabelian) finite simple group and$p$is a prime such that the$p$-Sylow in$G$is abelian. What can be said about its normalizer? I'm particularly interested in lower bounds on ... 4 votes 1 answer 569 views ### Schreier conjecture -- without a simple proof? and sporadic simple groups The Schreier conjecture asserts that$\mathrm{Out}(G)$is always a solvable group when$G$is a finite simple group. This result is known to be true as a corollary of the classification of finite ... 19 votes 1 answer 398 views ### Sporadic subgroup of E7 The dimensions of some representations of the Janko group J1 coinside with dimensions of smallest representations of the Lie algebra of type E7 (56, 133). It seems to be natural that there is a ... 14 votes 0 answers 693 views ### Algebra for the Baby I am reading the following article. Ryba, Alexander J.E., A natural invariant algebra for the Baby Monster group., J. Group Theory 10, No. 1, 55-69 (2007). ZBL1228.20012.. Author works with 4370-... 13 votes 3 answers 940 views ### Construction of representations of the Mathieu groups? The Mathieu groups are beautiful simple finite groups. (They were the first sporadic groups to be discovered in 1861-1870). They are related with many other miraculous constructions in mathematics: ... 5 votes 0 answers 487 views ### Atlas of finite groups, Character table of automorphism group of sporadic group I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group. I am reading from Inverse Galois Theory by G. Malle Let me start with$G=M_{12}$This(image ... 9 votes 2 answers 502 views ### Strongly real elements of odd order in sporadic finite simple groups Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ... 35 votes 2 answers 1k views ### Why do sporadic simple groups have so few conjugacy classes? In finite group theory, there's a general intuition that the further away a group is from abelian, the fewer conjugacy classes it will have. So it is to be expected that non-abelian finite simple ... 15 votes 0 answers 1k views ### How many sporadic simple groups are there, really? I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ... 19 votes 2 answers 1k views ### (weak?) BN-Pair / Tits System for Sporadic Groups The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial$N$and ... 5 votes 1 answer 362 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\$ ... 