Difference between revisions of "Linear representation theory of groups of order 81"

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information type = linear representation theory|
 
information type = linear representation theory|
 
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{{quotation|To understand these in a broader context, see [[linear representation theory of groups of prime-fourth order]] <nowiki>|</nowiki> [[linear representation theory of groups of order 3^n]]}}
  
 
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| [[Elementary abelian group:E81]] || 15 || 1 || [[linear representation theory of elementary abelian group:E81]], [[linear representation theory of elementary abelian groups]]
 
| [[Elementary abelian group:E81]] || 15 || 1 || [[linear representation theory of elementary abelian group:E81]], [[linear representation theory of elementary abelian groups]]
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==Degrees of irreducible representations==
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{{irrep degrees facts to check against}}
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{| class="sortable" border="1"
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! Group  !! GAP ID second part !! Nilpotency class !! Degrees as list !! Number of irreps of degree 1 ([[number of one-dimensional representations equals order of abelianization|= order of abelianization]]) !! Number of irreps of degree <math>p</math> !! Total number of irreps ([[number of irreducible representations equals number of conjugacy classes|= number of conjugacy classes]])
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|-
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| [[Cyclic group:Z81]] || 1 || 1 || 1 (81 times) || 81 || 0 || 81
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|-
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| [[Direct product of Z9 and Z9]] || 2 || 1 || 1 (81 times) || 81 || 0 || 81
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|-
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| [[SmallGroup(81,3)]] || 3 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33
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|-
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| [[Nontrivial semidirect product of Z9 and Z9]] || 4 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33
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|-
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| [[Direct product of Z27 and Z3]] || 5 || 1 || 1 (81 times) || 81 || 0 || 81
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|-
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| [[Semidirect product of Z27 and Z3]] || 6 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33
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|-
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| [[Wreath product of Z3 and Z3]] || 7 || 3 || 1 (9 times), 3 (8 times) || 9 || 8 || 17
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|-
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| [[SmallGroup(81,8)]] || 8 || 3 || 1 (9 times), 3 (8 times) || 9 || 8 || 17
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|-
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| [[SmallGroup(81,9)]] || 9 || 3 || 1 (9 times), 3 (8 times) || 9 || 8 || 17
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|-
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| [[SmallGroup(81,10)]] || 10 || 3 || 1 (9 times), 3 (8 times) || 9 || 8 || 17
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|-
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| [[Direct product of Z9 and E9]] || 11 || 1 || 1 (81 times) || 81 || 0 || 81
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|-
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| [[Direct product of prime-cube order group:U(3,3) and Z3]] || 12 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33
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|-
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| [[Direct product of semidirect product of Z9 and Z3 and Z3]] || 13 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33semidirect product of Z9 and Z3 and Z3]]
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|-
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| [[Central product of prime-cube order group:U(3,3) and Z9]] || 14 || 2 || 1 (27 times), 3 (6 times) || 27 || 6 || 33
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|-
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| [[Elementary abelian group:E81]] || 15 || 1 || 1 (81 times) || 81 || 0 || 81
 
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|}

Revision as of 01:18, 23 November 2011

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 81.
View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 81
To understand these in a broader context, see linear representation theory of groups of prime-fourth order | linear representation theory of groups of order 3^n
Common name for group Second part of GAP ID (GAP ID is (p^4, second part)) Nilpotency class Linear representation theory page
Cyclic group:Z81 1 1 linear representation theory of cyclic group:Z81, linear representation theory of cyclic groups
Direct product of Z9 and Z9 2 1 linear representation theory of direct product of Z9 and E9, linear representation theory of finite abelian groups
SmallGroup(81,3) 3 2 linear representation theory of SmallGroup(81,3)
Nontrivial semidirect product of Z9 and Z9 4 2 linear representation theory of nontrivial semidirect product of Z9 and Z9
Direct product of Z27 and Z3 5 1 linear representation theory of direct product of Z27 and Z3, linear representation theory of finite abelian groups
Semidirect product of Z27 and Z3 6 2 linear representation theory of semidirect product of Z27 and Z3
Wreath product of Z3 and Z3 7 3 linear representation theory of wreath product of Z3 and Z3
SmallGroup(81,8) 8 3 linear representation theory of SmallGroup(81,8)
SmallGroup(81,9) 9 3 linear representation theory of SmallGroup(81,9)
SmallGroup(81,10) 10 3 linear representation theory of SmallGroup(81,10)
Direct product of Z9 and E9 11 1 linear representation theory of direct product of Z9 and E9, linear representation theory of finite abelian groups
Direct product of prime-cube order group:U(3,3) and Z3 12 2 linear representation theory of direct product of prime-cube order group:U(3,3) and Z3
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 linear representation theory of direct product of semidirect product of Z9 and Z3 and Z3
Central product of prime-cube order group:U(3,3) and Z9 14 2 linear representation theory of central product of prime-cube order group:U(3,3) and Z9
Elementary abelian group:E81 15 1 linear representation theory of elementary abelian group:E81, linear representation theory of elementary abelian groups

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
Group GAP ID second part Nilpotency class Degrees as list Number of irreps of degree 1 (= order of abelianization) Number of irreps of degree p Total number of irreps (= number of conjugacy classes)
Cyclic group:Z81 1 1 1 (81 times) 81 0 81
Direct product of Z9 and Z9 2 1 1 (81 times) 81 0 81
SmallGroup(81,3) 3 2 1 (27 times), 3 (6 times) 27 6 33
Nontrivial semidirect product of Z9 and Z9 4 2 1 (27 times), 3 (6 times) 27 6 33
Direct product of Z27 and Z3 5 1 1 (81 times) 81 0 81
Semidirect product of Z27 and Z3 6 2 1 (27 times), 3 (6 times) 27 6 33
Wreath product of Z3 and Z3 7 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,8) 8 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,9) 9 3 1 (9 times), 3 (8 times) 9 8 17
SmallGroup(81,10) 10 3 1 (9 times), 3 (8 times) 9 8 17
Direct product of Z9 and E9 11 1 1 (81 times) 81 0 81
Direct product of prime-cube order group:U(3,3) and Z3 12 2 1 (27 times), 3 (6 times) 27 6 33
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 1 (27 times), 3 (6 times) 27 6 33semidirect product of Z9 and Z3 and Z3]]
Central product of prime-cube order group:U(3,3) and Z9 14 2 1 (27 times), 3 (6 times) 27 6 33
Elementary abelian group:E81 15 1 1 (81 times) 81 0 81