Element structure of groups of order 243: Difference between revisions

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| [[Elementary abelian group:E243]] || 1 || 242 || 0 || 0 || 0 || 0
| [[Elementary abelian group:E243]] || 1 || 242 || 0 || 0 || 0 || 0
|}
|}
|}
 
Here is the GAP code to generate these order statistics:<toggledisplay>
 
<tt>List([1..67],i ->[OrderStatistics(SmallGroup(243,i)),i]);</tt>
 
after first defining the [[GAP:OrderStatistics|OrderStatistics]] function (follow link for GAP code for function definition). The output is:
 
<pre>[ [ [ 1, 2, 6, 18, 54, 162 ], 1 ], [ [ 1, 26, 216, 0, 0, 0 ], 2 ], [ [ 1, 134, 108, 0, 0, 0 ], 3 ], [ [ 1, 80, 162, 0, 0, 0 ], 4 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 5 ], [ [ 1, 80, 162, 0, 0, 0 ], 6 ], [ [ 1, 26, 216, 0, 0, 0 ], 7 ], [ [ 1, 26, 216, 0, 0, 0 ], 8 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 9 ], [ [ 1, 8, 72, 162, 0, 0 ], 10 ], [ [ 1, 8, 72, 162, 0, 0 ], 11 ], [ [ 1, 26, 54, 162, 0, 0 ], 12 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 13 ], [ [ 1, 26, 216, 0, 0, 0 ], 14 ], [ [ 1, 26, 216, 0, 0, 0 ], 15 ], [ [ 1, 26, 54, 162, 0, 0 ], 16 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 17 ], [ [ 1, 26, 216, 0, 0, 0 ], 18 ], [ [ 1, 26, 54, 162, 0, 0 ], 19 ], [ [ 1, 26, 54, 162, 0, 0 ], 20 ],
  [ [ 1, 8, 72, 162, 0, 0 ], 21 ], [ [ 1, 8, 72, 162, 0, 0 ], 22 ], [ [ 1, 8, 18, 54, 162, 0 ], 23 ], [ [ 1, 8, 18, 54, 162, 0 ], 24 ],
  [ [ 1, 62, 180, 0, 0, 0 ], 25 ], [ [ 1, 170, 72, 0, 0, 0 ], 26 ], [ [ 1, 8, 234, 0, 0, 0 ], 27 ], [ [ 1, 116, 126, 0, 0, 0 ], 28 ],
  [ [ 1, 8, 234, 0, 0, 0 ], 29 ], [ [ 1, 62, 180, 0, 0, 0 ], 30 ], [ [ 1, 26, 216, 0, 0, 0 ], 31 ], [ [ 1, 80, 162, 0, 0, 0 ], 32 ],
  [ [ 1, 26, 216, 0, 0, 0 ], 33 ], [ [ 1, 26, 216, 0, 0, 0 ], 34 ], [ [ 1, 80, 162, 0, 0, 0 ], 35 ], [ [ 1, 26, 216, 0, 0, 0 ], 36 ],
  [ [ 1, 242, 0, 0, 0, 0 ], 37 ], [ [ 1, 80, 162, 0, 0, 0 ], 38 ], [ [ 1, 80, 162, 0, 0, 0 ], 39 ], [ [ 1, 80, 162, 0, 0, 0 ], 40 ],
  [ [ 1, 26, 216, 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, 54, 162, 0, 0 ], 48 ],
  [ [ 1, 26, 54, 162, 0, 0 ], 49 ], [ [ 1, 26, 54, 162, 0, 0 ], 50 ], [ [ 1, 134, 108, 0, 0, 0 ], 51 ], [ [ 1, 80, 162, 0, 0, 0 ], 52 ],
  [ [ 1, 188, 54, 0, 0, 0 ], 53 ], [ [ 1, 26, 216, 0, 0, 0 ], 54 ], [ [ 1, 80, 162, 0, 0, 0 ], 55 ], [ [ 1, 134, 108, 0, 0, 0 ], 56 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 57 ], [ [ 1, 188, 54, 0, 0, 0 ], 58 ], [ [ 1, 26, 216, 0, 0, 0 ], 59 ], [ [ 1, 80, 162, 0, 0, 0 ], 60 ],
  [ [ 1, 80, 162, 0, 0, 0 ], 61 ], [ [ 1, 242, 0, 0, 0, 0 ], 62 ], [ [ 1, 80, 162, 0, 0, 0 ], 63 ], [ [ 1, 80, 162, 0, 0, 0 ], 64 ],
  [ [ 1, 242, 0, 0, 0, 0 ], 65 ], [ [ 1, 80, 162, 0, 0, 0 ], 66 ], [ [ 1, 242, 0, 0, 0, 0 ], 67 ] ]</pre></toggledisplay>

Revision as of 01:58, 2 July 2010

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

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
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 1 242 0 0 0 0

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

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