# Linear representation theory of groups of order 36

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 36.

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## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroupSize bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroupCumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

### Grouping by degrees of irreducible representations

Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 | Number of irreps of degree 4 | Total number of irreps = number of conjugacy classes | Number of groups with these degrees of irreps | Nilpotency class(es) attained | Derived lengths attained | Description of groups | List of groups | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|

36 | 0 | 0 | 0 | 36 | 4 | 1 | 1 | abelian groups | 2, 5, 8, 14 | |

12 | 6 | 0 | 0 | 18 | 2 | not nilpotent | 2 | 6, 12 | ||

9 | 0 | 3 | 0 | 12 | 2 | not nilpotent | 2 | 3, 11 | ||

4 | 0 | 0 | 2 | 6 | 1 | not nilpotent | 2 | 9 | ||

4 | 8 | 0 | 0 | 12 | 4 | not nilpotent | 2 | 1, 4, 7, 13 | ||

4 | 4 | 0 | 1 | 9 | 1 | not nilpotent | 2 | 10 |