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Properties

Label 12T131
12T131 1 4 1->4 2 5 2->5 3 6 3->6 7 4->7 8 4->8 5->8 9 6->9 10 7->10 11 8->11 12 8->12 9->12 10->1 11->2 12->3 12->4
Degree $12$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\wr C_4$

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Copy content magma:G := TransitiveGroup(12, 131);
 

Group invariants

Abstract group:  $C_3\wr C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $324=2^{2} \cdot 3^{4}$
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 $n$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $131$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[3^{4}]4=3wr4$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3^2:C_4$, $C_3\times (C_3 : C_4)$
$108$:  12T72, 12T73

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T131 x 3, 18T123 x 4, 36T497 x 4, 36T514 x 2, 36T527 x 2, 36T532 x 4, 36T536 x 4, 36T543 x 4, 36T544 x 8

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $9$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
3A1 $3^{4}$ $1$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3A-1 $3^{4}$ $1$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3B $3^{4}$ $2$ $3$ $8$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$
3C1 $3^{2},1^{6}$ $2$ $3$ $4$ $( 2,10, 6)( 4,12, 8)$
3C-1 $3^{2},1^{6}$ $2$ $3$ $4$ $( 2, 6,10)( 4, 8,12)$
3D $3^{4}$ $4$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
3E $3^{2},1^{6}$ $4$ $3$ $4$ $( 2,10, 6)( 4, 8,12)$
3F1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 1, 9, 5)( 4, 8,12)$
3F-1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 3,11, 7)( 4, 8,12)$
3G1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 1, 9, 5)( 2,10, 6)$
3G-1 $3^{2},1^{6}$ $4$ $3$ $4$ $( 3, 7,11)( 4, 8,12)$
3H1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)$
3H-1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 9, 5)( 2,10, 6)( 4,12, 8)$
3I1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 5, 9)( 3, 7,11)( 4,12, 8)$
3I-1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)$
3J1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 2, 6,10)( 3, 7,11)( 4,12, 8)$
3J-1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 9, 5)( 2,10, 6)( 3, 7,11)$
3K1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 5, 9)( 2, 6,10)( 4,12, 8)$
3K-1 $3^{3},1^{3}$ $4$ $3$ $6$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)$
3L1 $3^{4}$ $4$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4, 8,12)$
3L-1 $3^{4}$ $4$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4, 8,12)$
3M1 $3,1^{9}$ $4$ $3$ $2$ $( 3, 7,11)$
3M-1 $3,1^{9}$ $4$ $3$ $2$ $( 4,12, 8)$
4A1 $4^{3}$ $27$ $4$ $9$ $( 1, 6,11, 8)( 2, 7, 4, 9)( 3,12, 5,10)$
4A-1 $4^{3}$ $27$ $4$ $9$ $( 1, 8,11, 6)( 2, 9, 4, 7)( 3,10, 5,12)$
6A1 $6^{2}$ $9$ $6$ $10$ $( 1, 3, 9,11, 5, 7)( 2, 8,10, 4, 6,12)$
6A-1 $6^{2}$ $9$ $6$ $10$ $( 1, 7, 5,11, 9, 3)( 2,12, 6, 4,10, 8)$
6B $6^{2}$ $18$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2, 4, 6, 8,10,12)$
6C1 $6,2^{3}$ $18$ $6$ $8$ $( 1, 7)( 2,12,10, 8, 6, 4)( 3, 9)( 5,11)$
6C-1 $6,2^{3}$ $18$ $6$ $8$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$
12A1 $12$ $27$ $12$ $11$ $( 1, 4, 3, 6, 9,12,11, 2, 5, 8, 7,10)$
12A-1 $12$ $27$ $12$ $11$ $( 1,10, 7, 8, 5, 2,11,12, 9, 6, 3, 4)$
12A5 $12$ $27$ $12$ $11$ $( 1,12, 7, 6, 5, 4,11,10, 9, 8, 3, 2)$
12A-5 $12$ $27$ $12$ $11$ $( 1, 2, 3, 8, 9,10,11, 4, 5, 6, 7,12)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed