↑

Properties

Label 12T1
12T1 1 4 1->4 5 1->5 2 2->5 6 2->6 3 3->6 7 3->7 4->7 8 4->8 5->8 9 5->9 6->9 10 6->10 7->10 11 7->11 8->11 12 8->12 9->1 9->12 10->1 10->2 11->2 11->3 12->3 12->4
Degree $12$
Order $12$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{12}$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(12, 1);
 

Group invariants

Abstract group:  $C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $12=2^{2} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  yes
Copy content magma:IsCyclic(G);
 
Abelian:  yes
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $C(4)[x]C(3)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $12$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

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 $1^{12}$ $1$ $1$ $0$ $()$
2A $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
3A1 $3^{4}$ $1$ $3$ $8$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
3A-1 $3^{4}$ $1$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
4A1 $4^{3}$ $1$ $4$ $9$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
4A-1 $4^{3}$ $1$ $4$ $9$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
6A1 $6^{2}$ $1$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6A-1 $6^{2}$ $1$ $6$ $10$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$
12A1 $12$ $1$ $12$ $11$ $( 1, 8, 3,10, 5,12, 7, 2, 9, 4,11, 6)$
12A-1 $12$ $1$ $12$ $11$ $( 1, 6,11, 4, 9, 2, 7,12, 5,10, 3, 8)$
12A5 $12$ $1$ $12$ $11$ $( 1,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
12A-5 $12$ $1$ $12$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1
Type
12.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1
12.2.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
12.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
12.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
12.2.1d1 C 1 −1 1 1 −i i −1 −1 i −i i −i
12.2.1d2 C 1 −1 1 1 i −i −1 −1 −i i −i i
12.2.1e1 C 1 1 ζ3−1 ζ3 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
12.2.1e2 C 1 1 ζ3 ζ3−1 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
12.2.1f1 C 1 −1 −ζ122 ζ124 −ζ123 ζ123 −ζ124 ζ122 −ζ12 ζ125 −ζ125 ζ12
12.2.1f2 C 1 −1 ζ124 −ζ122 ζ123 −ζ123 ζ122 −ζ124 ζ125 −ζ12 ζ12 −ζ125
12.2.1f3 C 1 −1 −ζ122 ζ124 ζ123 −ζ123 −ζ124 ζ122 ζ12 −ζ125 ζ125 −ζ12
12.2.1f4 C 1 −1 ζ124 −ζ122 −ζ123 ζ123 ζ122 −ζ124 −ζ125 ζ12 −ζ12 ζ125

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{12} + \left(-24 t^{4} + 24 t^{2} - 24\right) x^{10} + \left(144 t^{8} - 216 t^{6} + 360 t^{4} - 216 t^{2} + 144\right) x^{8} + \left(-256 t^{12} + 96 t^{10} - 752 t^{6} + 528 t^{4} - 432 t^{2} - 64\right) x^{6} + \left(1536 t^{14} - 4080 t^{12} + 7008 t^{10} - 6288 t^{8} + 4128 t^{6} - 1200 t^{4} + 384 t^{2}\right) x^{4} + \left(-2304 t^{16} + 6912 t^{14} - 12096 t^{12} + 12672 t^{10} - 8640 t^{8} + 3456 t^{6} - 576 t^{4}\right) x^{2} + \left(1024 t^{18} - 3072 t^{16} + 4608 t^{14} - 4096 t^{12} + 2112 t^{10} - 576 t^{8} + 64 t^{6}\right)$ Copy content Toggle raw display