Groups of order 123

This article gives information about, and links to more details on, groups of order 123
See pages on algebraic structures of order 123 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 123. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 123
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 123
subgroup structure	subgroup structure of groups of order 123
linear representation theory	linear representation theory of groups of order 123 projective representation theory of groups of order 123 modular representation theory of groups of order 123
endomorphism structure, automorphism structure	endomorphism structure of groups of order 123
group cohomology	group cohomology of groups of order 123

Statistics at a glance

Quantity	Value	Explanation
Total number of groups	1	See classification of cyclicity-forcing numbers
Number of abelian groups	1	only one group
Number of simple groups	0	cyclic group of composite order
Number of nilpotent groups	1	See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.

There is, up to isomorphism, a unique group of order $123$ , namely cyclic group:Z123. That is, $123$ is a cyclicity-forcing number.