A table of finite groups

Core idea

Open the interactive finite-group table.

This table organizes finite groups by their constructions and their place in the classification of finite simple groups. Select a tile to see its order, defining parameters, simplicity conditions, and a linked definition.

Two ways to explore

The classification layout gives the finite simple groups a clear structure: prime-order cyclic and alternating families, classical and exceptional Lie-type families, and the sporadics. The Tits group occupies a separate boundary position beside the exceptional families.

The all-groups layout places familiar families and small examples alongside the classification entries. These include symmetric, dihedral, quaternion, abelian, matrix, and finite Heisenberg groups. A family tile specifies parameters; an individual tile specifies one group. Order sorting compares individual groups using exact integers and keeps symbolic families separate.

Exceptional families and sporadic groups

An exceptional Lie-type label such as $E_8(q)$ denotes a family varying with a finite field. A sporadic label such as $M_{11}$ denotes one isomorphism type. The distinction explains why ten exceptional-family tiles and 26 sporadic tiles have different meanings.

The sporadic layout follows four display groups: Mathieu, Leech-related, Monster-related, and pariahs. These headings organize the entries; proximity is not an assertion of a direct subgroup inclusion. Verified relationships appear separately in the catalogue explorer.

Beyond simple groups

The simple-group classification provides possible composition factors. It does not classify all ways of assembling a finite group from those factors. The finite abelian classification is a complete classification within a smaller class.

Broader properties cut across the table: being a p-group, nilpotent, or solvable does not specify a single group from an order parameter. Follow those definitions to explore the classes.

Maps and further reading

• Hom, End, and Aut for finite cyclic groups.
• Endomorphisms of finite simple groups.
• Sporadic groups: definitions and orders.
• Lie-type groups: parameters and small cases.
• Familiar finite groups and examples.
• The category of finite groups and the full subcategory of finite simple groups.
• Explore recorded maps and relationships · Full mathematical catalogue.

References

• R. A. Wilson and collaborators, ATLAS of Finite Group Representations: sporadic groups, display sections “Mathieu groups,” “Leech lattice groups,” “Monster sections,” and “Pariahs.”