Properties

Label 24T25
24T25 1 20 1->20 23 1->23 2 19 2->19 24 2->24 3 10 3->10 12 3->12 22 3->22 4 9 4->9 11 4->11 21 4->21 5 5->20 5->22 5->23 6 6->19 6->21 6->24 7 8 7->8 13 7->13 17 7->17 14 8->14 18 8->18 16 9->16 9->17 15 10->15 10->18 11->14 11->18 12->13 12->17 13->11 14->12 15->9 15->21 15->23 16->10 16->22 16->24 17->8 18->7 19->5 19->20 20->6 21->3 22->4 23->2 24->1
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6:D_4$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 25);
 
Copy content sage:G = TransitiveGroup(24, 25)
 
Copy content oscar:G = transitive_group(24, 25)
 
Copy content gap:G := TransitiveGroup(24, 25);
 

Group invariants

Abstract group:  $C_6:D_4$
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 $t$:  $25$
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);
 
$\card{\Aut(F/K)}$:  $4$
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:  $(3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20)$, $(1,23,2,24)(3,22,4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13)$, $(1,20)(2,19)(3,10)(4,9)(5,23)(6,24)(7,13)(8,14)(11,18)(12,17)(15,21)(16,22)$
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
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $D_4\times C_2$
$24$:  $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $D_{6}$ x 3

Degree 8: $D_4\times C_2$

Degree 12: $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2

Low degree siblings

24T25, 24T45 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 $1^{24}$ $1$ $1$ $0$ $()$
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 $2^{12}$ $1$ $2$ $12$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,22)(10,21)(11,24)(12,23)$
2C $2^{12}$ $1$ $2$ $12$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,21)(10,22)(11,23)(12,24)$
2D $2^{12}$ $2$ $2$ $12$ $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,12)( 6,11)( 9,16)(10,15)(13,19)(14,20)(17,23)(18,24)$
2E $2^{12}$ $2$ $2$ $12$ $( 1,20)( 2,19)( 3,10)( 4, 9)( 5,23)( 6,24)( 7,13)( 8,14)(11,18)(12,17)(15,21)(16,22)$
2F $2^{10},1^{4}$ $6$ $2$ $10$ $( 1,18)( 2,17)( 5,13)( 6,14)( 7,24)( 8,23)( 9,10)(11,19)(12,20)(21,22)$
2G $2^{12}$ $6$ $2$ $12$ $( 1,13)( 2,14)( 3,24)( 4,23)( 5,10)( 6, 9)( 7,19)( 8,20)(11,15)(12,16)(17,21)(18,22)$
3A $3^{8}$ $2$ $3$ $16$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15)( 8,23,16)$
4A $4^{6}$ $6$ $4$ $18$ $( 1,23, 2,24)( 3,22, 4,21)( 5,20, 6,19)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$
4B $4^{6}$ $6$ $4$ $18$ $( 1,20, 2,19)( 3,17, 4,18)( 5,16, 6,15)( 7,14, 8,13)( 9,12,10,11)(21,24,22,23)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,22,17,14, 9, 5)( 2,21,18,13,10, 6)( 3,24,19,15,11, 7)( 4,23,20,16,12, 8)$
6B $6^{4}$ $2$ $6$ $20$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
6C $6^{4}$ $2$ $6$ $20$ $( 1,21,17,13, 9, 6)( 2,22,18,14,10, 5)( 3,23,19,16,11, 8)( 4,24,20,15,12, 7)$
6D1 $6^{4}$ $2$ $6$ $20$ $( 1,23, 9, 8,17,16)( 2,24,10, 7,18,15)( 3,13,11,21,19, 6)( 4,14,12,22,20, 5)$
6D-1 $6^{4}$ $2$ $6$ $20$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
6E1 $6^{4}$ $2$ $6$ $20$ $( 1, 4,17,20, 9,12)( 2, 3,18,19,10,11)( 5, 8,22,23,14,16)( 6, 7,21,24,13,15)$
6E-1 $6^{4}$ $2$ $6$ $20$ $( 1, 3,17,19, 9,11)( 2, 4,18,20,10,12)( 5, 7,22,24,14,15)( 6, 8,21,23,13,16)$

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 2D 2E 2F 2G 3A 4A 4B 6A 6B 6C 6D1 6D-1 6E1 6E-1
Size 1 1 1 1 2 2 6 6 2 6 6 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 2A 2A 3A 3A 3A 3A 3A 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 4A 4B 2B 2A 2C 2D 2D 2E 2E
Type
48.43.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.43.1b R 1 1 −1 −1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1
48.43.1c R 1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 −1 −1 1 1
48.43.1d R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 1 1 −1 −1
48.43.1e R 1 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1
48.43.1f R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
48.43.1g R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1
48.43.1h R 1 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1
48.43.2a R 2 2 2 2 2 2 0 0 −1 0 0 −1 −1 −1 −1 −1 −1 −1
48.43.2b R 2 −2 −2 2 0 0 0 0 2 0 0 −2 −2 2 0 0 0 0
48.43.2c R 2 −2 2 −2 0 0 0 0 2 0 0 2 −2 −2 0 0 0 0
48.43.2d R 2 2 −2 −2 −2 2 0 0 −1 0 0 1 −1 1 1 1 −1 −1
48.43.2e R 2 2 −2 −2 2 −2 0 0 −1 0 0 1 −1 1 −1 −1 1 1
48.43.2f R 2 2 2 2 −2 −2 0 0 −1 0 0 −1 −1 −1 1 1 1 1
48.43.2g1 C 2 −2 −2 2 0 0 0 0 −1 0 0 1 1 −1 −1−2ζ3 1+2ζ3 −1−2ζ3 1+2ζ3
48.43.2g2 C 2 −2 −2 2 0 0 0 0 −1 0 0 1 1 −1 1+2ζ3 −1−2ζ3 1+2ζ3 −1−2ζ3
48.43.2h1 C 2 −2 2 −2 0 0 0 0 −1 0 0 −1 1 1 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3
48.43.2h2 C 2 −2 2 −2 0 0 0 0 −1 0 0 −1 1 1 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3

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