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Properties

Label 12T45
12T45 1 4 1->4 5 1->5 10 1->10 2 2->5 2->5 6 2->6 3 3->6 7 3->7 4->7 8 4->8 5->8 9 5->9 6->9 6->9 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 $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times S_4$

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Copy content magma:G := TransitiveGroup(12, 45);
 

Group invariants

Abstract group:  $C_3\times S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $72=2^{3} \cdot 3^{2}$
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 $t$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $S(4)[x]C(3)$
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:  $(1,10)(2,5)(6,9)$, $(1,5,9)(2,6,10)(3,7,11)(4,8,12)$, $(1,4,7,10)(2,5,8,11)(3,6,9,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: $S_4$

Degree 6: None

Low degree siblings

18T30, 18T33, 24T80, 24T84, 36T20, 36T52

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 $2^{6}$ $3$ $2$ $6$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2B $2^{3},1^{6}$ $6$ $2$ $3$ $( 3,12)( 4, 7)( 8,11)$
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 $3^{3},1^{3}$ $8$ $3$ $6$ $( 1, 7, 4)( 3,12, 9)( 5,11, 8)$
3C1 $3^{4}$ $8$ $3$ $8$ $( 1,11,12)( 2, 6,10)( 3, 4, 5)( 7, 8, 9)$
3C-1 $3^{4}$ $8$ $3$ $8$ $( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$
4A $4^{3}$ $6$ $4$ $9$ $( 1,10, 4, 7)( 2, 8,11, 5)( 3, 9, 6,12)$
6A1 $6^{2}$ $3$ $6$ $10$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$
6A-1 $6^{2}$ $3$ $6$ $10$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
6B1 $6,3^{2}$ $6$ $6$ $9$ $( 1, 5, 9)( 2, 6,10)( 3, 4,11,12, 7, 8)$
6B-1 $6,3^{2}$ $6$ $6$ $9$ $( 1, 9, 5)( 2,10, 6)( 3, 8, 7,12,11, 4)$
12A1 $12$ $6$ $12$ $11$ $( 1,11,12,10, 5, 3, 4, 2, 9, 7, 8, 6)$
12A-1 $12$ $6$ $12$ $11$ $( 1, 3, 8,10, 9,11, 4, 6, 5, 7,12, 2)$

Malle's constant $a(G)$:     $1/3$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 6A1 6A-1 6B1 6B-1 12A1 12A-1
Size 1 3 6 1 1 8 8 8 6 3 3 6 6 6 6
2 P 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2A 3A-1 3A1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 4A 2A 2A 2B 2B 4A 4A
Type
72.42.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.42.1b R 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
72.42.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
72.42.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
72.42.1d1 C 1 1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
72.42.1d2 C 1 1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
72.42.2a R 2 2 0 2 2 −1 −1 −1 0 2 2 0 0 0 0
72.42.2b1 C 2 2 0 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 2ζ3−1 2ζ3 0 0 0 0
72.42.2b2 C 2 2 0 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 2ζ3 2ζ3−1 0 0 0 0
72.42.3a R 3 −1 1 3 3 0 0 0 −1 −1 −1 1 1 −1 −1
72.42.3b R 3 −1 −1 3 3 0 0 0 1 −1 −1 −1 −1 1 1
72.42.3c1 C 3 −1 1 3ζ3−1 3ζ3 0 0 0 −1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
72.42.3c2 C 3 −1 1 3ζ3 3ζ3−1 0 0 0 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
72.42.3d1 C 3 −1 −1 3ζ3−1 3ζ3 0 0 0 1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
72.42.3d2 C 3 −1 −1 3ζ3 3ζ3−1 0 0 0 1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed