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The McLaughlin group is the sporadic group McL of order |McL| = 898128000 (1) = 2^7·3^6·5^3·7·11. (2) It is implemented in the Wolfram Language as McLaughlinGroupMcL[].

The symmetric group S_n of degree n is the group of all permutations on n symbols. S_n is therefore a permutation group of order n! and contains as subgroups every group of ...

A group in which the elements are square matrices, the group multiplication law is matrix multiplication, and the group inverse is simply the matrix inverse. Every matrix ...

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi ...

The number of elements in a group G, denoted |G|. If the order of a group is a finite number, the group is said to be a finite group. The order of an element g of a finite ...

The icosahedral group I_h is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product A_5×Z_2 of the alternating ...

The pair group of a group G is the group that acts on the 2-subsets of {1,...,p} whose permutations are induced by G. Pair groups can be calculated using PairGroup[g] in the ...

An abstract group is a group characterized only by its abstract properties and not by the particular representations chosen for elements. For example, there are two distinct ...

The projective general linear group PGL_n(q) is the group obtained from the general linear group GL_n(q) on factoring by the scalar matrices contained in that group.

The monster group is the highest order sporadic group M. It has group order |M| = (1) = (2) where the divisors are precisely the 15 supersingular primes (Ogg 1980). The ...