Properties

Label 24T11
24T11 1 6 1->6 7 1->7 8 1->8 2 5 2->5 2->7 2->8 3 3->5 15 3->15 21 3->21 4 4->6 16 4->16 22 4->22 11 5->11 12 6->12 7->11 8->12 9 9->15 9->21 24 9->24 10 10->16 10->22 23 10->23 11->21 12->22 13 17 13->17 19 13->19 20 13->20 14 18 14->18 14->19 14->20 15->17 16->18 17->24 18->23 19->24 20->23
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times D_6$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 11);
 
Copy content sage:G = TransitiveGroup(24, 11)
 
Copy content oscar:G = transitive_group(24, 11)
 
Copy content gap:G := TransitiveGroup(24, 11);
 

Group invariants

Abstract group:  $C_2\times D_6$
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:  $24=2^{3} \cdot 3$
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 $n$:  $24$
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 $t$:  $11$
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);
 
$\card{\Aut(F/K)}$:  $24$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(24).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(24), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(24), G));
 
Generators:  $(1,8)(2,7)(3,21)(4,22)(5,11)(6,12)(9,15)(10,16)(13,19)(14,20)(17,24)(18,23)$, $(1,7)(2,8)(3,5)(4,6)(9,24)(10,23)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18)$, $(1,6)(2,5)(3,15)(4,16)(7,11)(8,12)(9,21)(10,22)(13,17)(14,18)(19,24)(20,23)$
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 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 3: $S_3$

Degree 4: $C_2^2$ x 7

Degree 6: $S_3$, $D_{6}$ x 6

Degree 8: $C_2^3$

Degree 12: $D_6$ x 3, $S_3 \times C_2^2$ x 4

Low degree siblings

12T10 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{12}$ $1$ $2$ $12$ $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,11)( 6,12)( 9,15)(10,16)(13,19)(14,20)(17,24)(18,23)$
2B $2^{12}$ $1$ $2$ $12$ $( 1,19)( 2,20)( 3,10)( 4, 9)( 5,23)( 6,24)( 7,14)( 8,13)(11,18)(12,17)(15,22)(16,21)$
2C $2^{12}$ $1$ $2$ $12$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,22)(10,21)(11,23)(12,24)$
2D $2^{12}$ $3$ $2$ $12$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,24)(10,23)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18)$
2E $2^{12}$ $3$ $2$ $12$ $( 1,10)( 2, 9)( 3,19)( 4,20)( 5, 6)( 7,15)( 8,16)(11,12)(13,21)(14,22)(17,18)(23,24)$
2F $2^{12}$ $3$ $2$ $12$ $( 1,14)( 2,13)( 3,23)( 4,24)( 5,10)( 6, 9)( 7,19)( 8,20)(11,16)(12,15)(17,22)(18,21)$
2G $2^{12}$ $3$ $2$ $12$ $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,17)(12,18)(13,16)(14,15)$
3A $3^{8}$ $2$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,20,12)( 4,19,11)( 5,22,13)( 6,21,14)( 7,24,16)( 8,23,15)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,15,18, 8, 9,23)( 2,16,17, 7,10,24)( 3, 6,20,21,12,14)( 4, 5,19,22,11,13)$
6B $6^{4}$ $2$ $6$ $20$ $( 1,11, 9,19,18, 4)( 2,12,10,20,17, 3)( 5,15,13,23,22, 8)( 6,16,14,24,21, 7)$
6C $6^{4}$ $2$ $6$ $20$ $( 1,22,18,13, 9, 5)( 2,21,17,14,10, 6)( 3,24,20,16,12, 7)( 4,23,19,15,11, 8)$

Malle's constant $a(G)$:     $1/12$

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 2F 2G 3A 6A 6B 6C
Size 1 1 1 1 3 3 3 3 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3A 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 2A 2B 2C
Type
24.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1
24.14.1b R 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1
24.14.1c R 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1
24.14.1d R 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1
24.14.1e R 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1
24.14.1f R 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
24.14.1g R 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
24.14.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1
24.14.2a R 2 2 2 2 0 0 0 0 −1 −1 −1 −1
24.14.2b R 2 −2 −2 2 0 0 0 0 −1 1 1 −1
24.14.2c R 2 −2 2 −2 0 0 0 0 −1 1 −1 1
24.14.2d R 2 2 −2 −2 0 0 0 0 −1 −1 1 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