Element structure of groups of order 243: Difference between revisions

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===Grouping by cumulative conjugacy class sizes (number of elements)===
{| class="sortable" border="1"
! 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)
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| 243 || 243 || 243 || 243 || 243 || 7 || 1 || all the [[abelian group]]s 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 group]]s 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===
===Correspondence between degrees of irreducible representations and conjugacy class sizes===


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===Pairs where one of the groups is abelian===
===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 group]]s). All the groups of nilpotency class ''exactly'' two are 1-isomorphic to [[abelian group]]s 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:
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 group]]s). All the groups of nilpotency class ''exactly'' two are 1-isomorphic to [[abelian group]]s 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:


{| class="sortable" border="1"
{| class="sortable" border="1"

Latest revision as of 12:39, 6 July 2011

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 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

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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

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.
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 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:

Non-abelian member of pair Nilpotency class GAP ID second part Abelian member of pair GAP ID second part Nature of the 1-isomorphism
2 2 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 11 direct product of Z27 and Z9 10 Baer correspondence
2 12 direct product of Z27 and E9 48 Baer correspondence
2 21 direct product of Z27 and Z9 10 Baer correspondence
2 24 direct product of Z81 and Z3 23 Baer correspondence
2 32 direct product of Z9 and E27 61 Baer correspondence
2 33 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 34 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 35 direct product of Z9 and E27 61 Baer correspondence
2 36 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 37 elementary abelian group:E243 67 Baer correspondence
2 38 direct product of Z9 and E27 61 Baer correspondence
2 39 direct product of Z9 and E27 61 Baer correspondence
2 40 direct product of Z9 and E27 61 Baer correspondence
2 41 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 42 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 43 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 44 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 45 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 46 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 47 direct product of Z9 and Z9 and Z3 31 Baer correspondence
2 49 direct product of Z27 and E9 48 Baer correspondence
2 50 direct product of Z27 and E9 48 Baer correspondence
2 62 elementary abelian group:E243 67 Baer correspondence
2 63 direct product of Z9 and E27 61 Baer correspondence
2 64 direct product of Z9 and E27 61 Baer correspondence
2 65 elementary abelian group:E243 67 Baer correspondence
2 66 direct product of Z9 and E27 61 Baer correspondence
3 16 direct product of Z27 and E9 48 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
3 19 direct product of Z27 and E9 48 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
3 20 direct product of Z27 and E9 48 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
3 55 direct product of Z9 and E27 61 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
3 9 direct product of Z9 and Z9 and Z3 31 ?
3 14 direct product of Z9 and Z9 and Z3 31 ?
3 22 direct product of Z27 and Z9 10 ?
3 52 direct product of Z9 and E27 61 ?
3 57 direct product of Z9 and E27 61 ?
3 60 direct product of Z9 and E27 61 ?

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 is the GAP code to generate these order statistics:[SHOW MORE]

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

Here is the GAP code to generate these order statistics:[SHOW MORE]

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