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Properties

Label 12T58
Degree $12$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^3:A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 58);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $58$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:A_4$
CHM label:  $[2^{4}]6$
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:  (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(8,9), (1,12)(4,5)
magma: Generators(G);
 

Low degree resolvents

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

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

8T33 x 2, 12T58, 12T59 x 2, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389

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 $ $6$ $2$ $( 6, 7)(10,11)$
$ 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 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 4, 4, 2, 2 $ $12$ $4$ $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$
$ 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 $ $16$ $6$ $( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$

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:  96.70
magma: IdentifyGroup(G);
 
Character table:    
      2  5  4  4  5   1   1  3  3   1   1
      3  1  .  .  .   1   1  .  1   1   1

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

X.1      1  1  1  1   1   1  1  1   1   1
X.2      1  1  1  1  -1   1 -1 -1   1  -1
X.3      1  1  1  1   A -/A -1 -1  -A  /A
X.4      1  1  1  1  /A  -A -1 -1 -/A   A
X.5      1  1  1  1 -/A  -A  1  1 -/A  -A
X.6      1  1  1  1  -A -/A  1  1  -A -/A
X.7      3 -1 -1  3   .   . -1  3   .   .
X.8      3 -1 -1  3   .   .  1 -3   .   .
X.9      6 -2  2 -2   .   .  .  .   .   .
X.10     6  2 -2 -2   .   .  .  .   .   .

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

magma: CharacterTable(G);