# Linear representation theory of groups of order 243

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This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 243.
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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 subgroup
Size 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 subgroup
Cumulative 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 3 Number of irreps of degree 9 Total number of irreps = number of conjugacy classes Total number of groups Nilpotency class(es) attained by these Description of groups List of GAP IDs second part (ascending order)
243 0 0 243 7 1 All the abelian groups of order 243 1, 10, 23, 31, 48, 61, 67
81 18 0 99 15 2 2, 11, 12, 21, 24, 32, 33, 34, 35, 36, 49, 50, 62, 63, 64
81 0 2 83 2 2 the extraspecial groups of order 243 65, 66
27 24 0 51 24 2, 3 13, 14, 15, 16, 17, 18, 19, 20, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 54, 55
27 6 2 35 6 3 22, 56, 57, 58, 59, 60
9 26 0 35 10 3, 4 3, 4, 5, 6, 7, 8, 9, 25, 26, 27
9 8 2 19 3 4 28, 29, 30

### Correspondence between degrees of irreducible representations and conjugacy class sizes

For groups of order 243, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, though the converse does not hold, i.e., the degrees of irreducible representations need not determine the conjugacy class sizes. 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 3 Number of conjugacy classes of size 9 Number of conjugacy classes of size 27 Total number of conjugacy classes = number of irreducible representations Number of irreps of degree 1 Number of irreps of degree 3 Number of irreps of degree 9
243 0 0 0 243 243 0 0
27 72 0 0 99 81 18 0
3 80 0 0 83 81 0 2
9 24 18 0 51 27 24 0
9 0 26 0 35 9 26 0
3 26 0 6 35 9 26 0
3 8 24 0 35 27 6 2
3 2 8 6 19 9 8 2

Note that there are two possibilities for the conjugacy class size statistics corresponding to the degrees of irreducible representations with 9 of degree 1 and 26 of degree 3.