# Element structure of groups of order 243

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 243.

View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 243

To understand these in a broader context, see: element structure of groups of order 3^n|element structure of groups of prime-fifth order

## 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 centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Full listing

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### Grouping by conjugacy class sizes

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 | Total number of groups with these conjugacy class sizes | Nilpotency class(es) attained by these groups | Description of groups | List of GAP IDs second part (ascending order) |
---|---|---|---|---|---|---|---|---|

243 | 0 | 0 | 0 | 243 | 7 | 1 | all the abelian groups of order 243 | 1, 10, 23, 31, 48, 61, 67 |

27 | 72 | 0 | 0 | 99 | 15 | 2 | 2, 11, 12, 21, 24, 32, 33, 34, 35, 36, 49, 50, 62, 63, 64 | |

3 | 80 | 0 | 0 | 83 | 2 | 2 | the extraspecial groups of order 243 | 65, 66 |

9 | 24 | 18 | 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 | |

9 | 0 | 26 | 0 | 35 | 7 | 3 | 3, 4, 5, 6, 7, 8, 9 | |

3 | 26 | 0 | 6 | 35 | 3 | 4 | 25, 26, 27 | |

3 | 8 | 24 | 0 | 35 | 6 | 3 | 22, 56, 57, 57, 59, 60 | |

3 | 2 | 8 | 6 | 19 | 3 | 4 | 28, 29, 30 |

### Grouping by cumulative conjugacy class sizes (number of elements)

Number of elements in (size dividing 1) conjugacy classes | Number of elements in (size dividing 3) conjugacy classes | Number of elements in (size dividing 9) conjugacy classes | Number of elements in (size dividing 27) conjugacy classes | Total number of conjugacy classes | Total number of groups with these conjugacy class sizes | Nilpotency class(es) attained by these groups | Description of groups | List of GAP IDs second part (ascending order) |
---|---|---|---|---|---|---|---|---|

243 | 243 | 243 | 243 | 243 | 7 | 1 | all the abelian groups of order 243 | 1, 10, 23, 31, 48, 61, 67 |

27 | 243 | 243 | 243 | 99 | 15 | 2 | 2, 11, 12, 21, 24, 32, 33, 34, 35, 36, 49, 50, 62, 63, 64 | |

3 | 243 | 243 | 243 | 83 | 2 | 2 | the extraspecial groups of order 243 | 65, 66 |

9 | 81 | 243 | 243 | 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 | |

9 | 9 | 243 | 243 | 35 | 7 | 3 | 3, 4, 5, 6, 7, 8, 9 | |

3 | 81 | 81 | 243 | 35 | 3 | 4 | 25, 26, 27 | |

3 | 27 | 243 | 243 | 35 | 6 | 3 | 22, 56, 57, 57, 59, 60 | |

3 | 9 | 81 | 243 | 19 | 3 | 4 | 28, 29, 30 |

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

See also linear representation theory of groups of order 243#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.

### Facts illustrated by these listings

- Degrees of irreducible representations need not determine conjugacy class size statistics
- Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order
- Number of conjugacy classes need not determine conjugacy class size statistics for groups of prime-fifth order
- Conjugacy class size statistics need not determine nilpotency class for groups of prime-fifth order

## 1-isomorphism

### Pairs where one of the groups is abelian

Of the 67 groups of order 243, 7 are abelian, 28 have nilpotency class *exactly* two, 26 have nilpotency class *exactly* three, and 6 have nilpotency class *exactly* four (i.e., they are maximal class groups). All the groups of nilpotency class *exactly* two are 1-isomorphic to abelian groups by means of the Baer correspondence. There are 10 other examples arising from class three examples. Of these, four are explained partially. Here is summary information:

Nature of 1-isomorphism | Number of 1-isomorphisms between non-abelian and abelian group of this type | Number of 1-isomorphisms between non-abelian and abelian group of this nature, not of any of the preceding types | Note |
---|---|---|---|

Baer correspondence | 28 | 28 | Correspond precisely to the groups of nilpotency class exactly two, and there are 28 of these. |

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32 | 4 | The four new examples are: SmallGroup(243,16), SmallGroup(243,19), SmallGroup(243,20), and central product of Z9 and wreath product of Z3 and Z3 (ID: (243,55)) |

Below are the details:

### Grouping by abelian member

Below are listed, for each abelian group of order 243, the list of all groups of order 243 that are 1-isomorphic to it:

Abelian member | GAP ID second part | Total number of members (including abelian member) | GAP IDs second part for members that have nilpotency class exactly two | Number of members with class exactly two | GAP IDs second part for members that have nilpotency class exactly three | Number of members with class exactly three |
---|---|---|---|---|---|---|

direct product of Z27 and Z9 | 10 | 4 | 11, 21 | 2 | 22 | 1 |

direct product of Z81 and Z3 | 23 | 2 | 24 | 1 | 0 | |

direct product of Z9 and Z9 and Z3 | 31 | 14 | 2, 33, 34, 36, 41, 42, 43, 44, 45, 46, 47 | 11 | 9, 14 | 2 |

direct product of Z27 and E9 | 48 | 7 | 12, 49, 50 | 3 | 16, 19, 20 | 3 |

direct product of Z9 and E27 | 61 | 13 | 32, 35, 38, 39, 40, 63, 64, 66 | 8 | 52, 55, 57, 60 | 4 |

elementary abelian group:E243 | 67 | 4 | 37, 62, 65 | 3 | 0 | |

Total | -- | 44 | -- | 28 | -- | 10 |

## Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS(cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM(cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

### Order statistics raw data

Here are the order statistics (non-cumulative version):

Group | Second part of GAP ID | Order 1 | Order 3 | Order 9 | Order 27 | Order 81 | Order 243 |
---|---|---|---|---|---|---|---|

Cyclic group:Z243 | 1 | 1 | 2 | 6 | 18 | 54 | 162 |

2 | 1 | 26 | 216 | 0 | 0 | 0 | |

3 | 1 | 134 | 108 | 0 | 0 | 0 | |

4 | 1 | 80 | 162 | 0 | 0 | 0 | |

5 | 1 | 26 | 216 | 0 | 0 | 0 | |

6 | 1 | 80 | 162 | 0 | 0 | 0 | |

7 | 1 | 26 | 216 | 0 | 0 | 0 | |

8 | 1 | 26 | 216 | 0 | 0 | 0 | |

9 | 1 | 26 | 216 | 0 | 0 | 0 | |

Direct product of Z27 and Z9 | 10 | 1 | 8 | 72 | 162 | 0 | 0 |

11 | 1 | 8 | 72 | 162 | 0 | 0 | |

12 | 1 | 26 | 54 | 162 | 0 | 0 | |

13 | 1 | 80 | 162 | 0 | 0 | 0 | |

14 | 1 | 26 | 216 | 0 | 0 | 0 | |

15 | 1 | 26 | 216 | 0 | 0 | 0 | |

16 | 1 | 26 | 54 | 162 | 0 | 0 | |

17 | 1 | 80 | 162 | 0 | 0 | 0 | |

18 | 1 | 26 | 216 | 0 | 0 | 0 | |

19 | 1 | 26 | 54 | 162 | 0 | 0 | |

20 | 1 | 26 | 54 | 162 | 0 | 0 | |

21 | 1 | 8 | 72 | 162 | 0 | 0 | |

Sylow subgroup of holomorph of Z27 | 22 | 1 | 8 | 72 | 162 | 0 | 0 |

Direct product of Z81 and Z3 | 23 | 1 | 8 | 18 | 54 | 162 | 0 |

24 | 1 | 8 | 18 | 54 | 162 | 0 | |

25 | 1 | 62 | 180 | 0 | 0 | 0 | |

26 | 1 | 170 | 72 | 0 | 0 | 0 | |

27 | 1 | 8 | 234 | 0 | 0 | 0 | |

28 | 1 | 116 | 126 | 0 | 0 | 0 | |

29 | 1 | 8 | 234 | 0 | 0 | 0 | |

30 | 1 | 62 | 180 | 0 | 0 | 0 | |

Direct product of Z9 and Z9 and Z3 | 31 | 1 | 26 | 216 | 0 | 0 | 0 |

32 | 1 | 80 | 162 | 0 | 0 | 0 | |

33 | 1 | 26 | 216 | 0 | 0 | 0 | |

34 | 1 | 26 | 216 | 0 | 0 | 0 | |

35 | 1 | 80 | 162 | 0 | 0 | 0 | |

36 | 1 | 26 | 216 | 0 | 0 | 0 | |

37 | 1 | 242 | 0 | 0 | 0 | 0 | |

38 | 1 | 80 | 162 | 0 | 0 | 0 | |

39 | 1 | 80 | 162 | 0 | 0 | 0 | |

40 | 1 | 80 | 162 | 0 | 0 | 0 | |

41 | 1 | 26 | 216 | 0 | 0 | 0 | |

42 | 1 | 26 | 216 | 0 | 0 | 0 | |

43 | 1 | 26 | 216 | 0 | 0 | 0 | |

44 | 1 | 26 | 216 | 0 | 0 | 0 | |

45 | 1 | 26 | 216 | 0 | 0 | 0 | |

46 | 1 | 26 | 216 | 0 | 0 | 0 | |

47 | 1 | 26 | 216 | 0 | 0 | 0 | |

Direct product of Z27 and E9 | 48 | 1 | 26 | 54 | 162 | 0 | 0 |

49 | 1 | 26 | 54 | 162 | 0 | 0 | |

50 | 1 | 26 | 54 | 162 | 0 | 0 | |

51 | 1 | 134 | 108 | 0 | 0 | 0 | |

52 | 1 | 80 | 162 | 0 | 0 | 0 | |

53 | 1 | 188 | 54 | 0 | 0 | 0 | |

54 | 1 | 26 | 216 | 0 | 0 | 0 | |

55 | 1 | 80 | 162 | 0 | 0 | 0 | |

56 | 1 | 134 | 108 | 0 | 0 | 0 | |

57 | 1 | 80 | 162 | 0 | 0 | 0 | |

58 | 1 | 188 | 54 | 0 | 0 | 0 | |

59 | 1 | 26 | 216 | 0 | 0 | 0 | |

60 | 1 | 80 | 162 | 0 | 0 | 0 | |

Direct product of Z9 and E27 | 61 | 1 | 80 | 162 | 0 | 0 | 0 |

62 | 1 | 242 | 0 | 0 | 0 | 0 | |

63 | 1 | 80 | 162 | 0 | 0 | 0 | |

64 | 1 | 80 | 162 | 0 | 0 | 0 | |

65 | 1 | 242 | 0 | 0 | 0 | 0 | |

66 | 1 | 80 | 162 | 0 | 0 | 0 | |

Elementary abelian group:E243 | 67 | 1 | 242 | 0 | 0 | 0 | 0 |

Here are the order statistics (cumulative version):

Group | Second part of GAP ID | 1st roots | 3rd roots | 9th roots | 27th roots | 81st roots | 243th roots |
---|---|---|---|---|---|---|---|

1 | 1 | 3 | 9 | 27 | 81 | 243 | |

2 | 1 | 27 | 243 | 243 | 243 | 243 | |

3 | 1 | 135 | 243 | 243 | 243 | 243 | |

4 | 1 | 81 | 243 | 243 | 243 | 243 | |

5 | 1 | 27 | 243 | 243 | 243 | 243 | |

6 | 1 | 81 | 243 | 243 | 243 | 243 | |

7 | 1 | 27 | 243 | 243 | 243 | 243 | |

8 | 1 | 27 | 243 | 243 | 243 | 243 | |

9 | 1 | 27 | 243 | 243 | 243 | 243 | |

Direct product of Z27 and Z9 | 10 | 1 | 9 | 81 | 243 | 243 | 243 |

11 | 1 | 9 | 81 | 243 | 243 | 243 | |

12 | 1 | 27 | 81 | 243 | 243 | 243 | |

13 | 1 | 81 | 243 | 243 | 243 | 243 | |

14 | 1 | 27 | 243 | 243 | 243 | 243 | |

15 | 1 | 27 | 243 | 243 | 243 | 243 | |

16 | 1 | 27 | 81 | 243 | 243 | 243 | |

17 | 1 | 81 | 243 | 243 | 243 | 243 | |

18 | 1 | 27 | 243 | 243 | 243 | 243 | |

19 | 1 | 27 | 81 | 243 | 243 | 243 | |

20 | 1 | 27 | 81 | 243 | 243 | 243 | |

21 | 1 | 9 | 81 | 243 | 243 | 243 | |

Sylow subgroup of holomorph of Z27 | 22 | 1 | 9 | 81 | 243 | 243 | 243 |

Direct product of Z81 and Z3 | 23 | 1 | 9 | 27 | 81 | 243 | 243 |

24 | 1 | 9 | 27 | 81 | 243 | 243 | |

25 | 1 | 63 | 243 | 243 | 243 | 243 | |

26 | 1 | 171 | 243 | 243 | 243 | 243 | |

27 | 1 | 9 | 243 | 243 | 243 | 243 | |

28 | 1 | 117 | 243 | 243 | 243 | 243 | |

29 | 1 | 9 | 243 | 243 | 243 | 243 | |

30 | 1 | 63 | 243 | 243 | 243 | 243 | |

Direct product of Z9 and Z9 and Z3 | 31 | 1 | 27 | 243 | 243 | 243 | 243 |

32 | 1 | 81 | 243 | 243 | 243 | 243 | |

33 | 1 | 27 | 243 | 243 | 243 | 243 | |

34 | 1 | 27 | 243 | 243 | 243 | 243 | |

35 | 1 | 81 | 243 | 243 | 243 | 243 | |

36 | 1 | 27 | 243 | 243 | 243 | 243 | |

37 | 1 | 243 | 243 | 243 | 243 | 243 | |

38 | 1 | 81 | 243 | 243 | 243 | 243 | |

39 | 1 | 81 | 243 | 243 | 243 | 243 | |

40 | 1 | 81 | 243 | 243 | 243 | 243 | |

41 | 1 | 27 | 243 | 243 | 243 | 243 | |

42 | 1 | 27 | 243 | 243 | 243 | 243 | |

43 | 1 | 27 | 243 | 243 | 243 | 243 | |

44 | 1 | 27 | 243 | 243 | 243 | 243 | |

45 | 1 | 27 | 243 | 243 | 243 | 243 | |

46 | 1 | 27 | 243 | 243 | 243 | 243 | |

47 | 1 | 27 | 243 | 243 | 243 | 243 | |

Direct product of Z27 and E9 | 48 | 1 | 27 | 81 | 243 | 243 | 243 |

49 | 1 | 27 | 81 | 243 | 243 | 243 | |

50 | 1 | 27 | 81 | 243 | 243 | 243 | |

51 | 1 | 135 | 243 | 243 | 243 | 243 | |

52 | 1 | 81 | 243 | 243 | 243 | 243 | |

53 | 1 | 189 | 243 | 243 | 243 | 243 | |

54 | 1 | 27 | 243 | 243 | 243 | 243 | |

55 | 1 | 81 | 243 | 243 | 243 | 243 | |

56 | 1 | 135 | 243 | 243 | 243 | 243 | |

57 | 1 | 81 | 243 | 243 | 243 | 243 | |

58 | 1 | 189 | 243 | 243 | 243 | 243 | |

59 | 1 | 27 | 243 | 243 | 243 | 243 | |

60 | 1 | 81 | 243 | 243 | 243 | 243 | |

Direct product of Z9 and E27 | 61 | 1 | 81 | 243 | 243 | 243 | 243 |

62 | 1 | 243 | 243 | 243 | 243 | 243 | |

63 | 1 | 81 | 243 | 243 | 243 | 243 | |

64 | 1 | 81 | 243 | 243 | 243 | 243 | |

65 | 1 | 243 | 243 | 243 | 243 | 243 | |

66 | 1 | 81 | 243 | 243 | 243 | 243 | |

Elementary abelian group:E243 | 67 | 1 | 243 | 243 | 243 | 243 | 243 |