Properties

Label 24T34
24T34 1 6 1->6 8 1->8 2 5 2->5 7 2->7 3 3->6 4 4->5 23 7->23 24 8->24 9 21 9->21 9->23 10 22 10->22 10->24 11 19 11->19 11->21 12 20 12->20 12->22 13 17 13->17 13->19 14 18 14->18 14->20 15 16 15->16 15->18 16->17
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_{24}$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 34);
 
Copy content sage:G = TransitiveGroup(24, 34)
 
Copy content oscar:G = transitive_group(24, 34)
 
Copy content gap:G := TransitiveGroup(24, 34);
 

Group invariants

Abstract group:  $D_{24}$
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:  $48=2^{4} \cdot 3$
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 $t$:  $34$
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(24).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(24), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(24), G));
 
Generators:  $(1,8)(2,7)(3,6)(4,5)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,18)(16,17)$, $(1,6)(2,5)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)(15,16)$
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
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$16$:  $D_{8}$
$24$:  $D_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: $D_{8}$

Degree 12: $D_{12}$

Low degree siblings

24T34

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 $2^{12}$ $1$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B $2^{11},1^{2}$ $12$ $2$ $11$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5,10)( 6, 9)( 7, 8)(15,23)(16,24)(17,21)(18,22)$
2C $2^{12}$ $12$ $2$ $12$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,12)( 6,11)( 7,10)( 8, 9)(17,23)(18,24)(19,21)(20,22)$
3A $3^{8}$ $2$ $3$ $16$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24)( 8,16,23)$
4A $4^{6}$ $2$ $4$ $18$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
8A1 $8^{3}$ $2$ $8$ $21$ $( 1,19,13, 8, 2,20,14, 7)( 3,21,16,10, 4,22,15, 9)( 5,24,17,11, 6,23,18,12)$
8A3 $8^{3}$ $2$ $8$ $21$ $( 1, 8,14,19, 2, 7,13,20)( 3,10,15,21, 4, 9,16,22)( 5,11,18,24, 6,12,17,23)$
12A1 $12^{2}$ $2$ $12$ $22$ $( 1,22,18,13, 9, 5, 2,21,17,14,10, 6)( 3,24,20,16,11, 7, 4,23,19,15,12, 8)$
12A5 $12^{2}$ $2$ $12$ $22$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
24A1 $24$ $2$ $24$ $23$ $( 1,23,22,19,18,15,13,12, 9, 8, 5, 3, 2,24,21,20,17,16,14,11,10, 7, 6, 4)$
24A5 $24$ $2$ $24$ $23$ $( 1,15, 5,20,10,23,13, 3,17, 7,22,12, 2,16, 6,19, 9,24,14, 4,18, 8,21,11)$
24A7 $24$ $2$ $24$ $23$ $( 1,16, 5,19,10,24,13, 4,17, 8,22,11, 2,15, 6,20, 9,23,14, 3,18, 7,21,12)$
24A11 $24$ $2$ $24$ $23$ $( 1,24,22,20,18,16,13,11, 9, 7, 5, 4, 2,23,21,19,17,15,14,12,10, 8, 6, 3)$

Malle's constant $a(G)$:     $1/11$

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 3A 4A 6A 8A1 8A3 12A1 12A5 24A1 24A5 24A7 24A11
Size 1 1 12 12 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 2A 3A 4A 4A 6A 6A 12A1 12A5 12A5 12A1
3 P 1A 2A 2B 2C 1A 4A 2A 8A3 8A1 4A 4A 8A1 8A3 8A1 8A3
Type
48.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1
48.7.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
48.7.1d R 1 1 1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
48.7.2a R 2 2 0 0 −1 2 −1 2 2 −1 −1 −1 −1 −1 −1
48.7.2b R 2 2 0 0 2 −2 2 0 0 −2 −2 0 0 0 0
48.7.2c R 2 2 0 0 −1 2 −1 −2 −2 −1 −1 1 1 1 1
48.7.2d1 R 2 −2 0 0 2 0 −2 −ζ8−1−ζ8 ζ8−1+ζ8 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
48.7.2d2 R 2 −2 0 0 2 0 −2 ζ8−1+ζ8 −ζ8−1−ζ8 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
48.7.2e1 R 2 2 0 0 −1 −2 −1 0 0 1 1 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
48.7.2e2 R 2 2 0 0 −1 −2 −1 0 0 1 1 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
48.7.2f1 R 2 −2 0 0 −1 0 1 −ζ24−3−ζ243 ζ24−3+ζ243 −ζ24−2−ζ242 ζ24−2+ζ242 ζ24−5+ζ245 ζ24−1+ζ24 −ζ24−1−ζ24 −ζ24−5−ζ245
48.7.2f2 R 2 −2 0 0 −1 0 1 −ζ24−3−ζ243 ζ24−3+ζ243 ζ24−2+ζ242 −ζ24−2−ζ242 −ζ24−1−ζ24 −ζ24−5−ζ245 ζ24−5+ζ245 ζ24−1+ζ24
48.7.2f3 R 2 −2 0 0 −1 0 1 ζ24−3+ζ243 −ζ24−3−ζ243 −ζ24−2−ζ242 ζ24−2+ζ242 −ζ24−5−ζ245 −ζ24−1−ζ24 ζ24−1+ζ24 ζ24−5+ζ245
48.7.2f4 R 2 −2 0 0 −1 0 1 ζ24−3+ζ243 −ζ24−3−ζ243 ζ24−2+ζ242 −ζ24−2−ζ242 ζ24−1+ζ24 ζ24−5+ζ245 −ζ24−5−ζ245 −ζ24−1−ζ24

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