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Properties

Label 22T3
22T3 1 19 1->19 21 1->21 2 20 2->20 22 2->22 3 17 3->17 3->19 4 18 4->18 4->20 5 15 5->15 5->18 6 16 6->16 6->17 7 14 7->14 7->15 8 13 8->13 8->16 9 11 9->11 9->14 10 12 10->12 10->13 11->12
Degree $22$
Order $44$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_{22}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(22, 3);
 
Copy content sage:G = TransitiveGroup(22, 3)
 
Copy content oscar:G = transitive_group(22, 3)
 
Copy content gap:G := TransitiveGroup(22, 3);
 

Group invariants

Abstract group:  $D_{22}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $44=2^{2} \cdot 11$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $22$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $3$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(22).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(22), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(22), G));
 
Generators:  $(1,19)(2,20)(3,17)(4,18)(5,15)(6,16)(7,14)(8,13)(9,11)(10,12)$, $(1,21)(2,22)(3,19)(4,20)(5,18)(6,17)(7,15)(8,16)(9,14)(10,13)(11,12)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$22$:  $D_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $D_{11}$

Low degree siblings

22T3, 44T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{22}$ $1$ $1$ $0$ $()$
2A $2^{11}$ $1$ $2$ $11$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
2B $2^{10},1^{2}$ $11$ $2$ $10$ $( 3,21)( 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15)(10,16)(11,13)(12,14)$
2C $2^{11}$ $11$ $2$ $11$ $( 1, 5)( 2, 6)( 3, 4)( 7,21)( 8,22)( 9,19)(10,20)(11,18)(12,17)(13,16)(14,15)$
11A1 $11^{2}$ $2$ $11$ $20$ $( 1,16, 7,22,14, 6,19,12, 4,18,10)( 2,15, 8,21,13, 5,20,11, 3,17, 9)$
11A2 $11^{2}$ $2$ $11$ $20$ $( 1, 7,14,19, 4,10,16,22, 6,12,18)( 2, 8,13,20, 3, 9,15,21, 5,11,17)$
11A3 $11^{2}$ $2$ $11$ $20$ $( 1,22,19,18,16,14,12,10, 7, 6, 4)( 2,21,20,17,15,13,11, 9, 8, 5, 3)$
11A4 $11^{2}$ $2$ $11$ $20$ $( 1,14, 4,16, 6,18, 7,19,10,22,12)( 2,13, 3,15, 5,17, 8,20, 9,21,11)$
11A5 $11^{2}$ $2$ $11$ $20$ $( 1, 6,10,14,18,22, 4, 7,12,16,19)( 2, 5, 9,13,17,21, 3, 8,11,15,20)$
22A1 $22$ $2$ $22$ $21$ $( 1,20,16,11, 7, 3,22,17,14, 9, 6, 2,19,15,12, 8, 4,21,18,13,10, 5)$
22A3 $22$ $2$ $22$ $21$ $( 1,11,22, 9,19, 8,18, 5,16, 3,14, 2,12,21,10,20, 7,17, 6,15, 4,13)$
22A5 $22$ $2$ $22$ $21$ $( 1, 3, 6, 8,10,11,14,15,18,20,22, 2, 4, 5, 7, 9,12,13,16,17,19,21)$
22A7 $22$ $2$ $22$ $21$ $( 1,17,12, 5,22,15,10, 3,19,13, 7, 2,18,11, 6,21,16, 9, 4,20,14, 8)$
22A9 $22$ $2$ $22$ $21$ $( 1, 9,18, 3,12,20, 6,13,22, 8,16, 2,10,17, 4,11,19, 5,14,21, 7,15)$

Malle's constant $a(G)$:     $1/10$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 11A1 11A2 11A3 11A4 11A5 22A1 22A3 22A5 22A7 22A9
Size 1 1 11 11 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 11A2 11A4 11A5 11A3 11A1 11A1 11A3 11A5 11A4 11A2
11 P 1A 2A 2B 2C 11A5 11A1 11A4 11A2 11A3 22A5 22A7 22A3 22A9 22A1
Type
44.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
44.3.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1
44.3.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1
44.3.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1
44.3.2a1 R 2 2 0 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−5+ζ115
44.3.2a2 R 2 2 0 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−4+ζ114
44.3.2a3 R 2 2 0 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−3+ζ113
44.3.2a4 R 2 2 0 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−2+ζ112
44.3.2a5 R 2 2 0 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−1+ζ11
44.3.2b1 R 2 −2 0 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−5−ζ115
44.3.2b2 R 2 −2 0 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−4−ζ114
44.3.2b3 R 2 −2 0 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−3−ζ113
44.3.2b4 R 2 −2 0 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−2−ζ112
44.3.2b5 R 2 −2 0 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−1−ζ11

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed