Element structure of groups of order 243: Difference between revisions

Revision as of 21:56, 6 June 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

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

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]

List([1..67],i ->[OrderStatistics(SmallGroup(243,i)),i]);

after first defining the OrderStatistics function (follow link for GAP code for function definition). The output is:

[ [ [ 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 ] ]

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]

List([1..67],i ->[OrderStatisticsCumulative(SmallGroup(243,i)),i]);

after first defining the OrderStatisticsCumulative function (follow link for GAP code for function definition). The output is:

[ [ [ 1, 3, 9, 27, 81, 243 ], 1 ], [ [ 1, 27, 243, 243, 243, 243 ], 2 ], [ [ 1, 135, 243, 243, 243, 243 ], 3 ], [ [ 1, 81, 243, 243, 243, 243 ], 4 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 5 ], [ [ 1, 81, 243, 243, 243, 243 ], 6 ], [ [ 1, 27, 243, 243, 243, 243 ], 7 ], [ [ 1, 27, 243, 243, 243, 243 ], 8 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 9 ], [ [ 1, 9, 81, 243, 243, 243 ], 10 ], [ [ 1, 9, 81, 243, 243, 243 ], 11 ], [ [ 1, 27, 81, 243, 243, 243 ], 12 ],
  [ [ 1, 81, 243, 243, 243, 243 ], 13 ], [ [ 1, 27, 243, 243, 243, 243 ], 14 ], [ [ 1, 27, 243, 243, 243, 243 ], 15 ],
  [ [ 1, 27, 81, 243, 243, 243 ], 16 ], [ [ 1, 81, 243, 243, 243, 243 ], 17 ], [ [ 1, 27, 243, 243, 243, 243 ], 18 ], [ [ 1, 27, 81, 243, 243, 243 ], 19 ]
    , [ [ 1, 27, 81, 243, 243, 243 ], 20 ], [ [ 1, 9, 81, 243, 243, 243 ], 21 ], [ [ 1, 9, 81, 243, 243, 243 ], 22 ], [ [ 1, 9, 27, 81, 243, 243 ], 23 ],
  [ [ 1, 9, 27, 81, 243, 243 ], 24 ], [ [ 1, 63, 243, 243, 243, 243 ], 25 ], [ [ 1, 171, 243, 243, 243, 243 ], 26 ], [ [ 1, 9, 243, 243, 243, 243 ], 27 ],
  [ [ 1, 117, 243, 243, 243, 243 ], 28 ], [ [ 1, 9, 243, 243, 243, 243 ], 29 ], [ [ 1, 63, 243, 243, 243, 243 ], 30 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 31 ], [ [ 1, 81, 243, 243, 243, 243 ], 32 ], [ [ 1, 27, 243, 243, 243, 243 ], 33 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 34 ], [ [ 1, 81, 243, 243, 243, 243 ], 35 ], [ [ 1, 27, 243, 243, 243, 243 ], 36 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 37 ], [ [ 1, 81, 243, 243, 243, 243 ], 38 ], [ [ 1, 81, 243, 243, 243, 243 ], 39 ],
  [ [ 1, 81, 243, 243, 243, 243 ], 40 ], [ [ 1, 27, 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, 81, 243, 243, 243 ], 48 ], [ [ 1, 27, 81, 243, 243, 243 ], 49 ]
    , [ [ 1, 27, 81, 243, 243, 243 ], 50 ], [ [ 1, 135, 243, 243, 243, 243 ], 51 ], [ [ 1, 81, 243, 243, 243, 243 ], 52 ],
  [ [ 1, 189, 243, 243, 243, 243 ], 53 ], [ [ 1, 27, 243, 243, 243, 243 ], 54 ], [ [ 1, 81, 243, 243, 243, 243 ], 55 ],
  [ [ 1, 135, 243, 243, 243, 243 ], 56 ], [ [ 1, 81, 243, 243, 243, 243 ], 57 ], [ [ 1, 189, 243, 243, 243, 243 ], 58 ],
  [ [ 1, 27, 243, 243, 243, 243 ], 59 ], [ [ 1, 81, 243, 243, 243, 243 ], 60 ], [ [ 1, 81, 243, 243, 243, 243 ], 61 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 62 ], [ [ 1, 81, 243, 243, 243, 243 ], 63 ], [ [ 1, 81, 243, 243, 243, 243 ], 64 ],
  [ [ 1, 243, 243, 243, 243, 243 ], 65 ], [ [ 1, 81, 243, 243, 243, 243 ], 66 ], [ [ 1, 243, 243, 243, 243, 243 ], 67 ] ]

@@ Line 1: / Line 1: @@
+{{all given order groups-specific information|
+order = 243|
+information type = element structure|
+connective = of}}
+==Conjugacy class sizes==
+{{conjugacy class structure facts to check against}}
+===Full listing===
+{{fillin}}
+===Grouping by conjugacy class sizes===
+{| class="sortable" border="1"
+! 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 group]]s 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 group]]s 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
+|}
 ==Order statistics==