Properties

Label 12T87
Degree $12$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 87);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $87$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:A_4$
CHM label:  $[2^{5}]6$
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:  (1,12)(2,3), (1,3,5,7,9,11)(2,4,6,8,10,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 3
$48$:  $C_2^2 \times A_4$
$96$:  $C_2^4:C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

12T87, 12T88 x 2, 16T416 x 2, 24T441 x 2, 24T442 x 2, 24T443 x 4, 24T444 x 2, 24T445 x 2, 24T446 x 2, 24T447 x 2, 24T448 x 4, 24T449, 24T450, 24T451 x 4, 24T452 x 4, 32T2183 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4, 6, 8,10)( 3, 5, 7, 9,11,12)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 6 $ $16$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,10)( 5,11)( 7,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 6,12, 7)( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,11)( 5,10)( 7,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$
$ 6, 6 $ $16$ $6$ $( 1, 8, 5,12, 9, 4)( 2,10, 7, 3,11, 6)$
$ 6, 6 $ $16$ $6$ $( 1,10, 8, 6, 4, 2)( 3,12,11, 9, 7, 5)$
$ 6, 6 $ $16$ $6$ $( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $192=2^{6} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  192.1000
magma: IdentifyGroup(G);
 
Character table:   
      2  6  5  5  6  5  5  6   2   2   2   2  4  4  4  4   2   2   2   2  6
      3  1  .  .  .  .  .  .   1   1   1   1  1  .  .  1   1   1   1   1  1

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

X.1      1  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   1 -1
X.3      1 -1  1 -1  1 -1  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 -1   1   1  -1  -1  1
X.5      1 -1  1 -1  1 -1  1   A  -A -/A  /A -1  1 -1  1  -A   A  /A -/A -1
X.6      1 -1  1 -1  1 -1  1  /A -/A  -A   A -1  1 -1  1 -/A  /A   A  -A -1
X.7      1 -1  1 -1  1 -1  1 -/A  /A  -A   A  1 -1  1 -1 -/A  /A  -A   A -1
X.8      1 -1  1 -1  1 -1  1  -A   A -/A  /A  1 -1  1 -1  -A   A -/A  /A -1
X.9      1  1  1  1  1  1  1   A   A -/A -/A -1 -1 -1 -1  -A  -A  /A  /A  1
X.10     1  1  1  1  1  1  1  /A  /A  -A  -A -1 -1 -1 -1 -/A -/A   A   A  1
X.11     1  1  1  1  1  1  1 -/A -/A  -A  -A  1  1  1  1 -/A -/A  -A  -A  1
X.12     1  1  1  1  1  1  1  -A  -A -/A -/A  1  1  1  1  -A  -A -/A -/A  1
X.13     3 -1 -1  3 -1 -1  3   .   .   .   . -3  1  1 -3   .   .   .   .  3
X.14     3 -1 -1  3 -1 -1  3   .   .   .   .  3 -1 -1  3   .   .   .   .  3
X.15     3  1 -1 -3 -1  1  3   .   .   .   . -3 -1  1  3   .   .   .   . -3
X.16     3  1 -1 -3 -1  1  3   .   .   .   .  3  1 -1 -3   .   .   .   . -3
X.17     6 -2 -2  2  2  2 -2   .   .   .   .  .  .  .  .   .   .   .   . -6
X.18     6 -2  2 -2 -2  2 -2   .   .   .   .  .  .  .  .   .   .   .   .  6
X.19     6  2 -2 -2  2 -2 -2   .   .   .   .  .  .  .  .   .   .   .   .  6
X.20     6  2  2  2 -2 -2 -2   .   .   .   .  .  .  .  .   .   .   .   . -6

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3

magma: CharacterTable(G);