Element structure of groups of order 12

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This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 12.
View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 12

The list

Group Second part of GAP ID (GAP ID is (12,second part)) Element structure page Element structure page (family contexts)
dicyclic group:Dic12 1 element structure of dicyclic group:Dic12 element structure of dicyclic groups
cyclic group:Z12 2 element structure of cyclic group:Z12 element structure of cyclic groups
alternating group:A4 3 element structure of alternating group:A4 element structure of alternating groups
dihedral group:D12 4 element structure of dihedral group:D12 element structure of dihedral groups
direct product of Z6 and Z2 5 element structure of direct product of Z6 and Z2 element structure of finite abelian groups

Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Full listing

Group Second part of GAP ID (GAP ID is (12,second part)) List of conjugacy class sizes Number of conjugacy classes of size 1 (= order of center) Number of conjugacy classes of size 2 Number of conjugacy classes of size 3 Number of conjugacy classes of size 4 Total number of conjugacy classes
dicyclic group:Dic12 1 1,1,2,2,3,3 2 2 2 0 6
cyclic group:Z12 2 1,1,1,1,1,1,1,1,1,1,1,1 12 0 0 0 12
alternating group:A4 3 1,3,4,4 1 0 1 2 4
dihedral group:D12 4 1,1,2,2,3,3 2 2 2 0 6
direct product of Z6 and Z2 5 1,1,1,1,1,1,1,1,1,1,1,1 12 0 0 0 12

Grouping by conjugacy class sizes

Number of conjugacy classes of size 1 Number of conjugacy classes of size 2 Number of conjugacy classes of size 3 Number of conjugacy classes of size 4 Total number of conjugacy classes Number of groups with these conjugacy class size statistics Description of groups List of groups List of GAP IDs (second part)
12 0 0 0 12 2 the abelian groups cyclic group:Z12, direct product of Z6 and Z2 2, 5
2 2 2 0 6 2 the dihedral and dicyclic group dicyclic group:Dic12, dihedral group:D12 1, 4
1 0 1 2 4 1 the alternating group alternating group:A4 3

Correspondence between degrees of irreducible representations and conjugacy class sizes

See also linear representation theory of groups of order 12#Degrees of irreducible representations.

For groups of order 12, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, and vice versa. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Number of conjugacy classes of size 1 Number of conjugacy classes of size 2 Number of conjugacy classes of size 3 Number of conjugacy classes of size 4 Total number of conjugacy classes = number of irreducible representations Number of irreps of degree 1 Number of irreps of degree 2 Number of irreps of degree 3
12 0 0 0 12 12 0 0
2 2 2 0 6 4 2 0
1 0 1 2 4 3 0 1

Note that the phenomenon of the conjugacy class size statistics and degrees of irreducible representations determining one another is not true for all orders: