↑

Properties

Label 12T48
Degree $12$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times S_4$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(12, 48);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\times S_4$
CHM label:   $2S_{4}(6)[x]2=[1/4.2^{6}]S(3)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_4\times C_2$ x 2

Low degree siblings

12T48 x 11, 16T182 x 4, 24T125, 24T126 x 6, 24T150 x 3, 24T151 x 4, 24T152 x 4, 32T388

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle 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 $ $3$ $2$ $( 4,10)( 5,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 4)( 3, 5)( 8,10)( 9,11)$
$ 4, 4, 1, 1, 1, 1 $ $6$ $4$ $( 2, 4, 8,10)( 3, 5, 9,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 4,10)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4, 5)( 6, 9)( 7, 8)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 6, 6 $ $8$ $6$ $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$
$ 6, 6 $ $8$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5)(10,11)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 2, 7, 8)( 3, 6, 9,12)( 4,11)( 5,10)$
$ 6, 6 $ $8$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$
$ 4, 4, 2, 2 $ $6$ $4$ $( 1, 3, 7, 9)( 2, 6, 8,12)( 4,10)( 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 6)( 2, 3)( 4, 5)( 7,12)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 9)( 3, 8)( 4,11)( 5,10)( 7,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 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:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.226
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 4A 4B 4C 4D 6A 6B 6C
Size 1 1 1 1 3 3 3 3 6 6 6 6 8 6 6 6 6 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2E 2E 2E 2E 3A 3A 3A
3 P 1A 2C 2A 2B 2D 2E 2F 2G 2K 2J 2I 2H 1A 4B 4A 4D 4C 2A 2B 2C
Type
96.226.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.226.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1
96.226.1c R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1
96.226.1d R 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 1 −1 1 −1 1 −1 1 −1
96.226.1e R 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 1 1 −1 1 −1 −1 1 −1
96.226.1f R 1 1 −1 −1 −1 1 1 −1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1
96.226.1g R 1 1 −1 −1 −1 1 1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1
96.226.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 1 1
96.226.2a R 2 2 2 2 2 2 2 2 0 0 0 0 −1 0 0 0 0 −1 −1 −1
96.226.2b R 2 −2 −2 2 2 2 −2 −2 0 0 0 0 −1 0 0 0 0 1 1 −1
96.226.2c R 2 −2 2 −2 −2 2 −2 2 0 0 0 0 −1 0 0 0 0 1 −1 1
96.226.2d R 2 2 −2 −2 −2 2 2 −2 0 0 0 0 −1 0 0 0 0 −1 1 1
96.226.3a R 3 3 3 3 −1 −1 −1 −1 1 1 1 1 0 −1 −1 −1 −1 0 0 0
96.226.3b R 3 3 3 3 −1 −1 −1 −1 −1 −1 −1 −1 0 1 1 1 1 0 0 0
96.226.3c R 3 −3 −3 3 −1 −1 1 1 −1 −1 1 1 0 −1 −1 1 1 0 0 0
96.226.3d R 3 −3 −3 3 −1 −1 1 1 1 1 −1 −1 0 1 1 −1 −1 0 0 0
96.226.3e R 3 −3 3 −3 1 −1 1 −1 −1 1 −1 1 0 1 −1 1 −1 0 0 0
96.226.3f R 3 −3 3 −3 1 −1 1 −1 1 −1 1 −1 0 −1 1 −1 1 0 0 0
96.226.3g R 3 3 −3 −3 1 −1 −1 1 −1 1 1 −1 0 −1 1 1 −1 0 0 0
96.226.3h R 3 3 −3 −3 1 −1 −1 1 1 −1 −1 1 0 1 −1 −1 1 0 0 0

magma: CharacterTable(G);