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Properties

Label 12T37
Degree $12$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_6$

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Show commands: Magma

magma: G := TransitiveGroup(12, 37);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_6$
CHM label:   $[3^{2}:2]E(4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5)(2,10)(4,8)(7,11), (2,6,10)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
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
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Low degree siblings

12T37, 18T29 x 4, 24T73, 36T34 x 2, 36T40 x 4

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 $2^{6}$ $1$ $2$ $6$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
2B $2^{6}$ $3$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{6}$ $3$ $2$ $6$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
2D $2^{6}$ $3$ $2$ $6$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,12)(10,11)$
2E $2^{6}$ $3$ $2$ $6$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
2F $2^{6}$ $9$ $2$ $6$ $( 1, 7)( 2,12)( 3, 5)( 4,10)( 6, 8)( 9,11)$
2G $2^{4},1^{4}$ $9$ $2$ $4$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)$
3A $3^{4}$ $2$ $3$ $8$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
3B $3^{4}$ $2$ $3$ $8$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
3C $3^{2},1^{6}$ $4$ $3$ $4$ $( 2, 6,10)( 4, 8,12)$
6A $6^{2}$ $2$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
6B $6^{2}$ $2$ $6$ $10$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$
6C $6,2^{3}$ $4$ $6$ $8$ $( 1,11, 9, 7, 5, 3)( 2, 8)( 4,10)( 6,12)$
6D $6^{2}$ $6$ $6$ $10$ $( 1,10, 9, 2, 5, 6)( 3, 8,11,12, 7, 4)$
6E $6^{2}$ $6$ $6$ $10$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$
6F $6^{2}$ $6$ $6$ $10$ $( 1, 6, 9, 2, 5,10)( 3, 8,11, 4, 7,12)$
6G $6^{2}$ $6$ $6$ $10$ $( 1,12, 5, 8, 9, 4)( 2, 3,10, 7, 6,11)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  72.46
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3A 3A 3B
3 P 1A 2A 2D 2C 2E 2B 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1
72.46.1c R 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1
72.46.1d R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1
72.46.1e R 1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1
72.46.1f R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
72.46.1g R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 1 −1 1 1 −1
72.46.1h R 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 1
72.46.2a R 2 2 0 2 2 0 0 0 2 −1 −1 −1 2 −1 0 −1 −1 0
72.46.2b R 2 2 2 0 0 2 0 0 −1 2 −1 2 −1 −1 −1 0 0 −1
72.46.2c R 2 −2 −2 0 0 2 0 0 −1 2 −1 −2 1 1 1 0 0 −1
72.46.2d R 2 −2 0 −2 2 0 0 0 2 −1 −1 1 −2 1 0 1 −1 0
72.46.2e R 2 −2 0 2 −2 0 0 0 2 −1 −1 1 −2 1 0 −1 1 0
72.46.2f R 2 −2 2 0 0 −2 0 0 −1 2 −1 −2 1 1 −1 0 0 1
72.46.2g R 2 2 −2 0 0 −2 0 0 −1 2 −1 2 −1 −1 1 0 0 1
72.46.2h R 2 2 0 −2 −2 0 0 0 2 −1 −1 −1 2 −1 0 1 1 0
72.46.4a R 4 4 0 0 0 0 0 0 −2 −2 1 −2 −2 1 0 0 0 0
72.46.4b R 4 −4 0 0 0 0 0 0 −2 −2 1 2 2 −1 0 0 0 0

magma: CharacterTable(G);