Properties

Label 24T36
24T36 1 3 1->3 2 4 2->4 18 3->18 23 3->23 17 4->17 24 4->24 5 8 5->8 21 5->21 6 7 6->7 22 6->22 20 7->20 7->22 19 8->19 8->21 9 11 9->11 9->17 10 12 10->12 10->18 11->2 16 11->16 12->1 15 12->15 13 14 13->14 13->15 14->16 15->5 16->6 17->19 18->20 19->10 20->9 21->24 22->23 23->13 24->14
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $Q_8:S_3$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 36);
 
Copy content sage:G = TransitiveGroup(24, 36)
 
Copy content oscar:G = transitive_group(24, 36)
 
Copy content gap:G := TransitiveGroup(24, 36);
 

Group invariants

Abstract group:  $Q_8:S_3$
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 $t$:  $36$
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:  $(1,3,18,20,9,11,2,4,17,19,10,12)(5,8,21,24,14,16,6,7,22,23,13,15)$, $(3,23)(4,24)(5,21)(6,22)(7,20)(8,19)(9,17)(10,18)(11,16)(12,15)(13,14)$
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
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$16$:  $QD_{16}$
$24$:  $(C_6\times C_2):C_2$

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: $QD_{16}$

Degree 12: $(C_6\times C_2):C_2$

Low degree siblings

There are no siblings 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 $2^{11},1^{2}$ $12$ $2$ $11$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10)( 6, 9)(15,24)(16,23)(17,21)(18,22)(19,20)$
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 $4^{6}$ $2$ $4$ $18$ $( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
4B $4^{6}$ $4$ $4$ $18$ $( 1,20, 2,19)( 3, 9, 4,10)( 5,24, 6,23)( 7,13, 8,14)(11,17,12,18)(15,21,16,22)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
8A1 $8^{3}$ $6$ $8$ $21$ $( 1,19,13, 8, 2,20,14, 7)( 3, 6,16,18, 4, 5,15,17)( 9,11,21,23,10,12,22,24)$
8A-1 $8^{3}$ $6$ $8$ $21$ $( 1,20,13, 7, 2,19,14, 8)( 3, 5,16,17, 4, 6,15,18)( 9,12,21,24,10,11,22,23)$
12A $12^{2}$ $4$ $12$ $22$ $( 1, 6,10,14,17,21, 2, 5, 9,13,18,22)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
12B1 $12^{2}$ $4$ $12$ $22$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,16,13,24,22, 8, 6,15,14,23,21, 7)$
12B-1 $12^{2}$ $4$ $12$ $22$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$

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 3A 4A 4B 6A 8A1 8A-1 12A 12B1 12B-1
Size 1 1 12 2 2 4 2 6 6 4 4 4
2 P 1A 1A 1A 3A 2A 2A 3A 4A 4A 6A 6A 6A
3 P 1A 2A 2B 1A 4A 4B 2A 8A1 8A-1 4A 4B 4B
Type
48.17.1a R 1 1 1 1 1 1 1 1 1 1 1 1
48.17.1b R 1 1 −1 1 1 −1 1 1 1 1 −1 −1
48.17.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 1
48.17.1d R 1 1 1 1 1 −1 1 −1 −1 1 −1 −1
48.17.2a R 2 2 0 −1 2 2 −1 0 0 −1 −1 −1
48.17.2b R 2 2 0 2 −2 0 2 0 0 −2 0 0
48.17.2c R 2 2 0 −1 2 −2 −1 0 0 −1 1 1
48.17.2d1 C 2 −2 0 2 0 0 −2 −ζ8−ζ83 ζ8+ζ83 0 0 0
48.17.2d2 C 2 −2 0 2 0 0 −2 ζ8+ζ83 −ζ8−ζ83 0 0 0
48.17.2e1 C 2 2 0 −1 −2 0 −1 0 0 1 −1−2ζ3 1+2ζ3
48.17.2e2 C 2 2 0 −1 −2 0 −1 0 0 1 1+2ζ3 −1−2ζ3
48.17.4a R 4 −4 0 −2 0 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