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The crystallographic point groups are the point groups in which translational periodicity is required (the so-called crystallography restriction). There are 32 such groups, ...

A point group is a group of symmetry operations which all leave at least one point unmoved. Although an isolated object may have an arbitrary Schönflies symbol, the ...

A point is a 0-dimensional mathematical object which can be specified in n-dimensional space using an n-tuple (x_1, x_2, ..., x_n) consisting of n coordinates. In dimensions ...

In the plane, there are 17 lattice groups, eight of which are pure translation. In R^3, there are 32 point groups and 230 space groups. In R^4, there are 4783 space lattice ...

Two groups are isomorphic if the correspondence between them is one-to-one and the "multiplication" table is preserved. For example, the point groups C_2 and D_1 are ...

Two groups G and H are said to be isoclinic if there are isomorphisms G/Z(G)->H/Z(H) and G^'->H^', where Z(G) is the group center of the group, which identify the two ...

The point group C_1 is a group on a single element that is isomorphic to the trivial group. Its character table is given below. C_1 1 1 1

The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups possible in a crystal lattice with the ...

The finite simple groups of Lie-type. They include four families of linear simple groups: PSL(n,q) (the projective special linear group), PSU(n,q) (the projective special ...

The automorphism group Co_1 of the Leech lattice modulo a center of order two is called "the" Conway group. There are 15 exceptional conjugacy classes of the Conway group. ...