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Properties

Label 24T24
24T24 1 16 1->16 20 1->20 21 1->21 2 15 2->15 19 2->19 22 2->22 3 6 3->6 18 3->18 24 3->24 4 5 4->5 17 4->17 23 4->23 5->1 5->15 5->20 6->2 6->16 6->19 7 7->4 10 7->10 14 7->14 8 8->3 9 8->9 13 8->13 9->6 12 9->12 9->23 10->5 11 10->11 10->24 11->7 11->13 12->8 12->14 13->3 13->10 14->4 14->9 15->12 15->18 16->11 16->17 17->8 17->13 18->7 18->14 19->15 19->21 20->16 20->22 21->11 21->18 21->23 22->12 22->17 22->24 23->1 23->19 24->2 24->20
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{12}:C_2$

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Copy content magma:G := TransitiveGroup(24, 24);
 

Group invariants

Abstract group:  $D_{12}:C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $48=2^{4} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,20)(2,19)(3,18)(4,17)(5,15)(6,16)(7,14)(8,13)(9,12)(10,11)(21,23)(22,24)$, $(1,16,17,8,9,23)(2,15,18,7,10,24)(3,6,19,21,11,13)(4,5,20,22,12,14)$, $(1,21,18,14,9,6,2,22,17,13,10,5)(3,24,20,16,11,7,4,23,19,15,12,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$16$:  $Q_8:C_2$
$24$:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 8: $Q_8:C_2$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T19, 24T24

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 $2^{12}$ $2$ $2$ $12$ $( 1, 8)( 2, 7)( 3,21)( 4,22)( 5,12)( 6,11)( 9,16)(10,15)(13,19)(14,20)(17,23)(18,24)$
2C $2^{10},1^{4}$ $6$ $2$ $10$ $( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$
2D $2^{12}$ $6$ $2$ $12$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,24)(14,23)(15,21)(16,22)(17,19)(18,20)$
3A $3^{8}$ $2$ $3$ $16$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15)( 8,23,16)$
4A1 $4^{6}$ $1$ $4$ $18$ $( 1,13, 2,14)( 3,15, 4,16)( 5,17, 6,18)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$
4A-1 $4^{6}$ $1$ $4$ $18$ $( 1,14, 2,13)( 3,16, 4,15)( 5,18, 6,17)( 7,19, 8,20)( 9,22,10,21)(11,23,12,24)$
4B $4^{6}$ $2$ $4$ $18$ $( 1,19, 2,20)( 3,10, 4, 9)( 5,23, 6,24)( 7,14, 8,13)(11,18,12,17)(15,22,16,21)$
4C $4^{6}$ $6$ $4$ $18$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,21,12,22)(13,20,14,19)(15,18,16,17)$
4D $4^{6}$ $6$ $4$ $18$ $( 1, 5, 2, 6)( 3,15, 4,16)( 7,12, 8,11)( 9,22,10,21)(13,17,14,18)(19,24,20,23)$
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 $6^{4}$ $2$ $6$ $20$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
6B-1 $6^{4}$ $2$ $6$ $20$ $( 1,23, 9, 8,17,16)( 2,24,10, 7,18,15)( 3,13,11,21,19, 6)( 4,14,12,22,20, 5)$
12A1 $12^{2}$ $2$ $12$ $22$ $( 1,22,18,13, 9, 5, 2,21,17,14,10, 6)( 3,23,20,15,11, 8, 4,24,19,16,12, 7)$
12A-1 $12^{2}$ $2$ $12$ $22$ $( 1, 6,10,14,17,21, 2, 5, 9,13,18,22)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
12B1 $12^{2}$ $2$ $12$ $22$ $( 1, 4,18,19, 9,12, 2, 3,17,20,10,11)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
12B5 $12^{2}$ $2$ $12$ $22$ $( 1,12,10,19,17, 4, 2,11, 9,20,18, 3)( 5,15,13,23,22, 7, 6,16,14,24,21, 8)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 3A 4A1 4A-1 4B 4C 4D 6A 6B1 6B-1 12A1 12A-1 12B1 12B5
Size 1 1 2 6 6 2 1 1 2 6 6 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 3A 2A 2A 2A 2A 2A 3A 3A 3A 6A 6A 6A 6A
3 P 1A 2A 2B 2C 2D 1A 4A-1 4A1 4B 4C 4D 2A 2B 2B 4A1 4A-1 4B 4B
Type
48.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.37.1b R 1 1 −1 −1 −1 1 −1 −1 1 1 1 −1 −1 1 −1 −1 1 1
48.37.1c R 1 1 −1 −1 1 1 1 1 −1 1 −1 −1 −1 1 1 1 −1 −1
48.37.1d R 1 1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 1 1 1 −1 −1
48.37.1e R 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 1 1
48.37.1f R 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1 1
48.37.1g R 1 1 1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
48.37.1h R 1 1 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1
48.37.2a R 2 2 2 0 0 −1 2 2 2 0 0 −1 −1 −1 −1 −1 −1 −1
48.37.2b R 2 2 −2 0 0 −1 −2 −2 2 0 0 1 1 −1 1 1 −1 −1
48.37.2c R 2 2 −2 0 0 −1 2 2 −2 0 0 1 1 −1 −1 −1 1 1
48.37.2d R 2 2 2 0 0 −1 −2 −2 −2 0 0 −1 −1 −1 1 1 1 1
48.37.2e1 C 2 −2 0 0 0 2 −2i 2i 0 0 0 0 0 −2 2i −2i 0 0
48.37.2e2 C 2 −2 0 0 0 2 2i −2i 0 0 0 0 0 −2 −2i 2i 0 0
48.37.2f1 C 2 −2 0 0 0 −1 −2ζ123 2ζ123 0 0 0 1−2ζ122 −1+2ζ122 1 −ζ123 ζ123 ζ12−1+ζ12 −ζ12−1−ζ12
48.37.2f2 C 2 −2 0 0 0 −1 2ζ123 −2ζ123 0 0 0 −1+2ζ122 1−2ζ122 1 ζ123 −ζ123 ζ12−1+ζ12 −ζ12−1−ζ12
48.37.2f3 C 2 −2 0 0 0 −1 −2ζ123 2ζ123 0 0 0 −1+2ζ122 1−2ζ122 1 −ζ123 ζ123 −ζ12−1−ζ12 ζ12−1+ζ12
48.37.2f4 C 2 −2 0 0 0 −1 2ζ123 −2ζ123 0 0 0 1−2ζ122 −1+2ζ122 1 ζ123 −ζ123 −ζ12−1−ζ12 ζ12−1+ζ12

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Regular extensions

Data not computed