Properties

Label 12T115
Degree $12$
Order $192$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4^2:D_6$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 115);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $115$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2:D_6$
CHM label:  $[2^{3}]S_{4}(6)_{8}$
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)(6,7)(8,9), (1,12)(2,3)(4,5), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (2,4)(3,5)(6,7)(8,10)(9,11)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Low degree siblings

12T112 x 2, 12T113 x 2, 12T114 x 2, 12T115, 16T431, 24T530 x 2, 24T531 x 2, 24T532 x 2, 24T533 x 2, 24T534 x 2, 24T535, 24T536 x 2, 24T537 x 2, 24T538 x 2, 24T539, 24T540, 24T541, 32T2145, 32T2146 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, 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,5)(6,7)(8,9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 4)( 3, 5)( 6, 7)( 8,10)( 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $12$ $4$ $( 2, 4, 3, 5)( 8,11, 9,10)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 8, 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 2, 8)( 3, 9)( 4,11)( 5,10)( 6, 7)$
$ 4, 4, 2, 1, 1 $ $12$ $4$ $( 2,10, 3,11)( 4, 8, 5, 9)( 6, 7)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 4)( 3, 5,12)( 6, 8,11)( 7, 9,10)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 2, 4,12, 3, 5)( 6, 9,11)( 7, 8,10)$
$ 8, 2, 2 $ $24$ $8$ $( 1, 2, 6, 9,12, 3, 7, 8)( 4,10)( 5,11)$
$ 8, 4 $ $24$ $8$ $( 1, 2, 6, 8,12, 3, 7, 9)( 4,11, 5,10)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 6,12, 7)( 2, 3)( 4,11, 5,10)( 8, 9)$

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:  192.956
magma: IdentifyGroup(G);
 
Character table:   
      2  6  4  6  4  4  5  4  4  4  1  1  3  3  5
      3  1  1  .  .  .  .  .  .  .  1  1  .  .  .

        1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d
     2P 1a 1a 1a 1a 2b 2b 1a 2b 1a 3a 3a 4b 4d 2b
     3P 1a 2a 2b 2c 4a 4b 2d 4c 2e 1a 2a 8a 8b 4d
     5P 1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d
     7P 1a 2a 2b 2c 4a 4b 2d 4c 2e 3a 6a 8a 8b 4d

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

magma: CharacterTable(G);