Properties

Label 24T1320
24T1320 1 7 1->7 15 1->15 2 8 2->8 16 2->16 3 4 3->4 5 3->5 6 4->6 18 5->18 17 6->17 10 7->10 21 7->21 9 8->9 22 8->22 9->16 10->15 11 12 11->12 13 11->13 14 12->14 13->10 20 13->20 14->9 19 14->19 15->14 15->18 16->13 16->17 23 17->23 24 17->24 18->23 18->24 19->20 19->22 20->21 21->2 21->11 22->1 22->12 23->2 23->6 24->1 24->5
Degree $24$
Order $432$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_2\times \PU(3,2)$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(24, 1320);
 
Copy content sage:G = TransitiveGroup(24, 1320)
 
Copy content oscar:G = transitive_group(24, 1320)
 
Copy content gap:G := TransitiveGroup(24, 1320);
 

Group invariants

Abstract group:  $C_2\times \PU(3,2)$
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:  $432=2^{4} \cdot 3^{3}$
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 $t$:  $1320$
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:  $(1,7,21,2,8,22)(3,4)(5,18,23,6,17,24)(9,16,13,10,15,14)(11,12)(19,20)$, $(1,15,18,24)(2,16,17,23)(3,5)(4,6)(7,10)(8,9)(11,13,20,21)(12,14,19,22)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$, $\SL(2,3)$ x 2
$48$:  16T59
$216$:  $(C_3^2:Q_8):C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $A_4$

Degree 6: None

Degree 8: $A_4\times C_2$

Degree 12: 12T122

Low degree siblings

18T151, 24T1321

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 $2^{8},1^{8}$ $9$ $2$ $8$ $( 1, 9)( 2,10)( 3,19)( 4,20)( 7,16)( 8,15)(13,21)(14,22)$
2C $2^{12}$ $9$ $2$ $12$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,24)( 8,23)( 9,17)(10,18)(13,22)(14,21)(15,16)(19,20)$
3A $3^{6},1^{6}$ $8$ $3$ $12$ $( 1,18, 9)( 2,17,10)( 3,19,12)( 4,20,11)( 7,16,23)( 8,15,24)$
3B1 $3^{6},1^{6}$ $12$ $3$ $12$ $( 3, 7, 5)( 4, 8, 6)(11,15,13)(12,16,14)(19,23,22)(20,24,21)$
3B-1 $3^{6},1^{6}$ $12$ $3$ $12$ $( 3, 5, 7)( 4, 6, 8)(11,13,15)(12,14,16)(19,22,23)(20,21,24)$
3C1 $3^{8}$ $24$ $3$ $16$ $( 1, 9,18)( 2,10,17)( 3,23,22)( 4,24,21)( 5,12, 7)( 6,11, 8)(13,20,15)(14,19,16)$
3C-1 $3^{8}$ $24$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,22,23)( 4,21,24)( 5, 7,12)( 6, 8,11)(13,15,20)(14,16,19)$
4A $4^{4},2^{4}$ $54$ $4$ $16$ $( 1,19, 9, 3)( 2,20,10, 4)( 5,24)( 6,23)( 7,13,16,21)( 8,14,15,22)(11,17)(12,18)$
4B $4^{4},2^{4}$ $54$ $4$ $16$ $( 1,24, 9,15)( 2,23,10,16)( 3,22,12, 5)( 4,21,11, 6)( 7,17)( 8,18)(13,20)(14,19)$
6A $6^{3},2^{3}$ $8$ $6$ $18$ $( 1,10,18, 2, 9,17)( 3,11,19, 4,12,20)( 5, 6)( 7,24,16, 8,23,15)(13,14)(21,22)$
6B1 $6^{3},2^{3}$ $12$ $6$ $18$ $( 1, 2)( 3, 6, 7, 4, 5, 8)( 9,10)(11,14,15,12,13,16)(17,18)(19,21,23,20,22,24)$
6B-1 $6^{3},2^{3}$ $12$ $6$ $18$ $( 1, 2)( 3, 8, 5, 4, 7, 6)( 9,10)(11,16,13,12,15,14)(17,18)(19,24,22,20,23,21)$
6C1 $6^{4}$ $24$ $6$ $20$ $( 1,17, 9, 2,18,10)( 3,21,23, 4,22,24)( 5, 8,12, 6, 7,11)(13,16,20,14,15,19)$
6C-1 $6^{4}$ $24$ $6$ $20$ $( 1,10,18, 2, 9,17)( 3,24,22, 4,23,21)( 5,11, 7, 6,12, 8)(13,19,15,14,20,16)$
6D1 $6^{3},2^{3}$ $36$ $6$ $18$ $( 1,16, 6, 2,15, 5)( 3,11)( 4,12)( 7,21,17,24,14, 9)( 8,22,18,23,13,10)(19,20)$
6D-1 $6^{3},2^{3}$ $36$ $6$ $18$ $( 1, 5,15, 2, 6,16)( 3,11)( 4,12)( 7, 9,14,24,17,21)( 8,10,13,23,18,22)(19,20)$
6E1 $6^{2},3^{2},2^{2},1^{2}$ $36$ $6$ $16$ $( 1,11,13,18, 4, 6)( 2,12,14,17, 3, 5)( 7,16)( 8,15)( 9,20,21)(10,19,22)$
6E-1 $6^{2},3^{2},2^{2},1^{2}$ $36$ $6$ $16$ $( 1, 6, 4,18,13,11)( 2, 5, 3,17,14,12)( 7,16)( 8,15)( 9,21,20)(10,22,19)$

Malle's constant $a(G)$:     $1/8$

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 3B1 3B-1 3C1 3C-1 4A 4B 6A 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1
Size 1 1 9 9 8 12 12 24 24 54 54 8 12 12 24 24 36 36 36 36
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 2B 2B 3A 3B1 3B-1 3C1 3C-1 3B1 3B-1 3B-1 3B1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 4B 2A 2A 2A 2A 2A 2C 2C 2B 2B
Type
432.735.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
432.735.1b R 1 −1 1 −1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1
432.735.1c1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
432.735.1c2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
432.735.1d1 C 1 −1 1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 −1 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
432.735.1d2 C 1 −1 1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 −1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1
432.735.2a S 2 −2 −2 2 2 −1 −1 −1 −1 0 0 −2 1 1 1 1 −1 −1 1 1
432.735.2b S 2 2 −2 −2 2 −1 −1 −1 −1 0 0 2 −1 −1 −1 −1 1 1 1 1
432.735.2c1 C 2 −2 −2 2 2 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 −2 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1
432.735.2c2 C 2 −2 −2 2 2 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 −2 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
432.735.2d1 C 2 2 −2 −2 2 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 2 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3 ζ3−1
432.735.2d2 C 2 2 −2 −2 2 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 2 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
432.735.3a R 3 3 3 3 3 0 0 0 0 −1 −1 3 0 0 0 0 0 0 0 0
432.735.3b R 3 −3 3 −3 3 0 0 0 0 1 −1 −3 0 0 0 0 0 0 0 0
432.735.8a R 8 8 0 0 −1 2 2 −1 −1 0 0 −1 2 2 −1 −1 0 0 0 0
432.735.8b R 8 −8 0 0 −1 2 2 −1 −1 0 0 1 −2 −2 1 1 0 0 0 0
432.735.8c1 C 8 8 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0 0 0
432.735.8c2 C 8 8 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0 0 0
432.735.8d1 C 8 −8 0 0 −1 2ζ3−1 2ζ3 −ζ3−1 −ζ3 0 0 1 −2ζ3 −2ζ3−1 ζ3 ζ3−1 0 0 0 0
432.735.8d2 C 8 −8 0 0 −1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 0 0 1 −2ζ3−1 −2ζ3 ζ3−1 ζ3 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