Properties

Label 24T37
24T37 1 15 1->15 16 1->16 2 2->15 2->16 3 3->1 5 3->5 4 4->2 6 4->6 11 5->11 19 5->19 12 6->12 20 6->20 7 9 7->9 22 7->22 8 10 8->10 21 8->21 9->8 24 9->24 10->7 23 10->23 14 11->14 17 11->17 13 12->13 18 12->18 13->4 13->4 14->3 14->3 15->14 15->17 16->13 16->18 17->7 17->23 18->8 18->24 19->9 19->22 20->10 20->21 21->12 21->20 22->11 22->19 23->2 23->6 24->1 24->5
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3:D_8$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 37);
 
Copy content sage:G = TransitiveGroup(24, 37)
 
Copy content oscar:G = transitive_group(24, 37)
 
Copy content gap:G := TransitiveGroup(24, 37);
 

Group invariants

Abstract group:  $C_3:D_8$
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 $t$:  $37$
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:  $(1,16,13,4,2,15,14,3)(5,11,17,23,6,12,18,24)(7,22,19,9,8,21,20,10)$, $(1,15,17,7,9,24)(2,16,18,8,10,23)(3,5,19,22,11,14)(4,6,20,21,12,13)$
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
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$16$:  $D_{8}$
$24$:  $(C_6\times C_2):C_2$

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: $(C_6\times C_2):C_2$

Low degree siblings

24T43

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 $2^{12}$ $4$ $2$ $12$ $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,12)( 6,11)( 9,16)(10,15)(13,19)(14,20)(17,23)(18,24)$
2C $2^{11},1^{2}$ $12$ $2$ $11$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5, 9)( 6,10)( 7, 8)(15,23)(16,24)(17,22)(18,21)$
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 $4^{6}$ $2$ $4$ $18$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
6B1 $6^{4}$ $4$ $6$ $20$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
6B-1 $6^{4}$ $4$ $6$ $20$ $( 1,23, 9, 8,17,16)( 2,24,10, 7,18,15)( 3,13,11,21,19, 6)( 4,14,12,22,20, 5)$
8A1 $8^{3}$ $6$ $8$ $21$ $( 1, 8,13,20, 2, 7,14,19)( 3,17,16, 6, 4,18,15, 5)( 9,23,21,12,10,24,22,11)$
8A3 $8^{3}$ $6$ $8$ $21$ $( 1, 7,13,19, 2, 8,14,20)( 3,18,16, 5, 4,17,15, 6)( 9,24,21,11,10,23,22,12)$
12A $12^{2}$ $4$ $12$ $22$ $( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,23,20,15,11, 8, 4,24,19,16,12, 7)$

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 6B1 6B-1 8A1 8A3 12A
Size 1 1 4 12 2 2 2 4 4 6 6 4
2 P 1A 1A 1A 1A 3A 2A 3A 3A 3A 4A 4A 6A
3 P 1A 2A 2B 2C 1A 4A 2A 2B 2B 8A3 8A1 4A
Type
48.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1
48.15.1b R 1 1 −1 −1 1 1 1 −1 −1 1 1 1
48.15.1c R 1 1 −1 1 1 1 1 −1 −1 −1 −1 1
48.15.1d R 1 1 1 −1 1 1 1 1 1 −1 −1 1
48.15.2a R 2 2 2 0 −1 2 −1 −1 −1 0 0 −1
48.15.2b R 2 2 0 0 2 −2 2 0 0 0 0 −2
48.15.2c R 2 2 −2 0 −1 2 −1 1 1 0 0 −1
48.15.2d1 R 2 −2 0 0 2 0 −2 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 0
48.15.2d2 R 2 −2 0 0 2 0 −2 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 0
48.15.2e1 C 2 2 0 0 −1 −2 −1 −1−2ζ3 1+2ζ3 0 0 1
48.15.2e2 C 2 2 0 0 −1 −2 −1 1+2ζ3 −1−2ζ3 0 0 1
48.15.4a R 4 −4 0 0 −2 0 2 0 0 0 0 0

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