Properties

Label 12T50
Degree $12$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\GL(2,\mathbb{Z}/4)$

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Show commands: Magma

magma: G := TransitiveGroup(12, 50);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(2,\mathbb{Z}/4)$
CHM label:  $1/2e[1/16.D(4)^{3}]S(3)$
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:  (3,9)(6,12), (1,7)(3,9)(5,11), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$
$12$:  $D_{6}$
$24$:  $S_4$, $(C_6\times C_2):C_2$
$48$:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T49 x 2, 12T52, 16T186, 16T193, 24T153, 24T154, 24T155 x 2, 24T156, 24T157, 24T158, 24T159, 24T165, 24T166, 32T392

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.195
magma: IdentifyGroup(G);
 
Character table:   
      2  5  4  5  4  5  3  3  3  3  2  2  2  2  5
      3  1  .  .  1  .  .  .  .  .  1  1  1  1  1

        1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6b 3a 6c 2f
     2P 1a 1a 1a 1a 1a 2b 1a 2f 2d 3a 3a 3a 3a 1a
     3P 1a 2a 2b 2c 2d 4a 2e 4b 4c 2f 2c 1a 2c 2f
     5P 1a 2a 2b 2c 2d 4a 2e 4b 4c 6a 6c 3a 6b 2f

X.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
X.3      1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1  1
X.4      1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1
X.5      2 -2  2 -2  2  .  .  .  . -1  1 -1  1  2
X.6      2  2  2  2  2  .  .  .  . -1 -1 -1 -1  2
X.7      2  . -2  .  2  .  .  .  . -2  .  2  . -2
X.8      2  . -2  .  2  .  .  .  .  1  A -1 -A -2
X.9      2  . -2  .  2  .  .  .  .  1 -A -1  A -2
X.10     3 -1 -1  3 -1 -1  1  1 -1  .  .  .  .  3
X.11     3 -1 -1  3 -1  1 -1 -1  1  .  .  .  .  3
X.12     3  1 -1 -3 -1 -1 -1  1  1  .  .  .  .  3
X.13     3  1 -1 -3 -1  1  1 -1 -1  .  .  .  .  3
X.14     6  .  2  . -2  .  .  .  .  .  .  .  . -6

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3

magma: CharacterTable(G);