↑

Properties

Label 12T123
12T123 1 3 1->3 11 1->11 12 1->12 2 2->3 4 2->4 2->4 5 3->5 3->5 4->5 6 4->6 7 5->7 6->7 8 6->8 6->8 9 7->9 7->9 8->9 8->12 9->1 10 10->11 10->12 12->2
Degree $12$
Order $240$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times S_5$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(12, 123);
 
Copy content sage:G = TransitiveGroup(12, 123)
 
Copy content oscar:G = transitive_group(12, 123)
 
Copy content gap:G := TransitiveGroup(12, 123);
 

Group invariants

Abstract group:  $C_2\times S_5$
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:  $240=2^{4} \cdot 3 \cdot 5$
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:  no
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 $n$:  $12$
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 $t$:  $123$
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);
 
CHM label:   $L(6):2[x]2$
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(12).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(12), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(12), G));
 
Generators:  $(1,3,5,7,9)(2,4,6,8,12)$, $(1,11)(2,4)(3,5)(6,8)(7,9)(10,12)$, $(1,12)(2,3)(4,5)(6,7)(8,9)(10,11)$
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
$4$:  $C_2^2$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $\PGL(2,5)$

Low degree siblings

10T22 x 2, 12T123, 20T62 x 2, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
2B $2^{6}$ $10$ $2$ $6$ $( 1, 9)( 2, 4)( 3, 5)( 6,10)( 7,11)( 8,12)$
2C $2^{6}$ $10$ $2$ $6$ $( 1, 8)( 2,11)( 3,10)( 4, 7)( 5, 6)( 9,12)$
2D $2^{6}$ $15$ $2$ $6$ $( 1, 4)( 2, 3)( 5,12)( 6, 9)( 7, 8)(10,11)$
2E $2^{4},1^{4}$ $15$ $2$ $4$ $( 1, 5)( 2,10)( 3,11)( 4,12)$
3A $3^{4}$ $20$ $3$ $8$ $( 1, 3, 7)( 2, 6,12)( 4,10, 8)( 5,11, 9)$
4A $4^{2},2^{2}$ $30$ $4$ $8$ $( 1, 2, 5,10)( 3, 4,11,12)( 6, 7)( 8, 9)$
4B $4^{2},1^{4}$ $30$ $4$ $6$ $( 1, 9,11, 3)( 2,12, 8,10)$
5A $5^{2},1^{2}$ $24$ $5$ $8$ $( 1, 9, 3,11, 7)( 2,10, 6,12, 8)$
6A $6^{2}$ $20$ $6$ $10$ $( 1,11, 3, 9, 7, 5)( 2, 8, 6, 4,12,10)$
6B $6^{2}$ $20$ $6$ $10$ $( 1, 8,11,12, 9,10)( 2, 5, 6, 3, 4, 7)$
6C $6^{2}$ $20$ $6$ $10$ $( 1, 6,11, 8, 5, 2)( 3,12, 7,10, 9, 4)$
10A $10,2$ $24$ $10$ $10$ $( 1,10, 9, 6, 3,12,11, 8, 7, 2)( 4, 5)$

Malle's constant $a(G)$:     $1/4$

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 2D 2E 3A 4A 4B 5A 6A 6B 6C 10A
Size 1 1 10 10 15 15 20 30 30 24 20 20 20 24
2 P 1A 1A 1A 1A 1A 1A 3A 2E 2E 5A 3A 3A 3A 5A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 5A 2B 2A 2C 10A
5 P 1A 2A 2B 2C 2D 2E 3A 4A 4B 1A 6A 6B 6C 2A
Type
240.189.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
240.189.1b R 1 −1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1
240.189.1c R 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1
240.189.1d R 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 −1 1
240.189.4a R 4 4 2 2 0 0 1 0 0 −1 −1 1 −1 −1
240.189.4b R 4 4 −2 −2 0 0 1 0 0 −1 1 1 1 −1
240.189.4c R 4 −4 −2 2 0 0 1 0 0 −1 1 −1 −1 1
240.189.4d R 4 −4 2 −2 0 0 1 0 0 −1 −1 −1 1 1
240.189.5a R 5 5 −1 −1 1 1 −1 1 1 0 −1 −1 −1 0
240.189.5b R 5 5 1 1 1 1 −1 −1 −1 0 1 −1 1 0
240.189.5c R 5 −5 −1 1 −1 1 −1 −1 1 0 −1 1 1 0
240.189.5d R 5 −5 1 −1 −1 1 −1 1 −1 0 1 1 −1 0
240.189.6a R 6 6 0 0 −2 −2 0 0 0 1 0 0 0 1
240.189.6b R 6 −6 0 0 2 −2 0 0 0 1 0 0 0 −1

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