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Properties

Label 12T73
Degree $12$
Order $108$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(12, 73);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $73$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_{12}$
CHM label:  $1/2[3^{3}:2]4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,7,2)(3,6,9,12)(4,11,10,5), (2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T73, 18T44 x 2, 27T33, 36T81 x 2, 36T95 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 3,11, 7)( 4, 8,12)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 1, 1, 1 $ $4$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 12 $ $9$ $12$ $( 1, 2, 3, 4, 5,10, 7,12, 9, 6,11, 8)$
$ 12 $ $9$ $12$ $( 1, 2, 3, 8, 9, 6,11,12, 5,10, 7, 4)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 2, 3,12)( 4, 9, 6,11)( 5,10, 7, 8)$
$ 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $9$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4, 6, 8,10,12)$
$ 12 $ $9$ $12$ $( 1, 4, 7,10, 5,12,11, 6, 9, 8, 3, 2)$
$ 12 $ $9$ $12$ $( 1, 4, 3, 6, 9, 8,11,10, 5,12, 7, 2)$
$ 4, 4, 4 $ $9$ $4$ $( 1, 4,11, 2)( 3,10, 5,12)( 6, 9, 8, 7)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 3 $ $1$ $3$ $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  108.36
magma: IdentifyGroup(G);
 
Character table:    
      2  2  .  .  .  .  .   2   2  2  2   2   2   2   2  2  .  2  2
      3  3  3  3  3  3  3   1   1  1  1   1   1   1   1  1  3  3  3

        1a 3a 3b 3c 3d 3e 12a 12b 4a 2a  6a  6b 12c 12d 4b 3f 3g 3h
     2P 1a 3b 3a 3c 3e 3d  6a  6b 2a 1a  3g  3h  6a  6b 2a 3f 3h 3g
     3P 1a 1a 1a 1a 1a 1a  4b  4b 4b 2a  2a  2a  4a  4a 4a 1a 1a 1a
     5P 1a 3b 3a 3c 3e 3d 12b 12a 4a 2a  6b  6a 12d 12c 4b 3f 3h 3g
     7P 1a 3a 3b 3c 3d 3e 12c 12d 4b 2a  6a  6b 12a 12b 4a 3f 3g 3h
    11P 1a 3b 3a 3c 3e 3d 12d 12c 4b 2a  6b  6a 12b 12a 4a 3f 3h 3g

X.1      1  1  1  1  1  1   1   1  1  1   1   1   1   1  1  1  1  1
X.2      1  1  1  1  1  1  -1  -1 -1  1   1   1  -1  -1 -1  1  1  1
X.3      1  1  1  1  1  1   C   C  C -1  -1  -1  -C  -C -C  1  1  1
X.4      1  1  1  1  1  1  -C  -C -C -1  -1  -1   C   C  C  1  1  1
X.5      1  A /A  1  A /A -/A  -A -1  1   A  /A -/A  -A -1  1 /A  A
X.6      1 /A  A  1 /A  A  -A -/A -1  1  /A   A  -A -/A -1  1  A /A
X.7      1  A /A  1  A /A  /A   A  1  1   A  /A  /A   A  1  1 /A  A
X.8      1 /A  A  1 /A  A   A  /A  1  1  /A   A   A  /A  1  1  A /A
X.9      1  A /A  1  A /A   D -/D  C -1  -A -/A  -D  /D -C  1 /A  A
X.10     1  A /A  1  A /A  -D  /D -C -1  -A -/A   D -/D  C  1 /A  A
X.11     1 /A  A  1 /A  A -/D   D  C -1 -/A  -A  /D  -D -C  1  A /A
X.12     1 /A  A  1 /A  A  /D  -D -C -1 -/A  -A -/D   D  C  1  A /A
X.13     4 -2 -2  1  1  1   .   .  .  .   .   .   .   .  . -2  4  4
X.14     4  1  1 -2 -2 -2   .   .  .  .   .   .   .   .  .  1  4  4
X.15     4  B /B  1 /A  A   .   .  .  .   .   .   .   .  . -2  E /E
X.16     4 /B  B  1  A /A   .   .  .  .   .   .   .   .  . -2 /E  E
X.17     4 /A  A -2  B /B   .   .  .  .   .   .   .   .  .  1  E /E
X.18     4  A /A -2 /B  B   .   .  .  .   .   .   .   .  .  1 /E  E

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -2*E(3)
  = 1-Sqrt(-3) = 1-i3
C = -E(4)
  = -Sqrt(-1) = -i
D = -E(12)^7
E = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3

magma: CharacterTable(G);