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Properties

Label 12T264
Degree $12$
Order $5184$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\wr C_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 264);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $264$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\wr C_4$
CHM label:   $[S(3)^{4}]4=S(3)wr4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (4,8), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $
$64$:  $((C_8 : C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T262, 18T482, 24T7730, 24T7731, 24T7732, 24T7733, 24T7734, 24T7735, 24T7736, 24T7744, 24T7745 x 2, 24T7746 x 2, 24T7747, 24T7748, 24T7749 x 2, 24T7750 x 2, 24T7751, 24T7752, 24T7753, 24T7754, 24T7786, 24T7787, 36T6231, 36T6232, 36T6233, 36T6234, 36T6235, 36T6236, 36T6237, 36T6269, 36T6291

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{12}$ $1$ $1$ $()$
$3,1^{9}$ $8$ $3$ $( 4,12, 8)$
$3^{2},1^{6}$ $16$ $3$ $( 3,11, 7)( 4,12, 8)$
$3^{2},1^{6}$ $8$ $3$ $( 2,10, 6)( 4,12, 8)$
$3^{3},1^{3}$ $32$ $3$ $( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$3^{4}$ $16$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$2,1^{10}$ $12$ $2$ $( 8,12)$
$3,2,1^{7}$ $24$ $6$ $( 3,11, 7)( 8,12)$
$3,2,1^{7}$ $24$ $6$ $( 2,10, 6)( 8,12)$
$3^{2},2,1^{4}$ $48$ $6$ $( 2,10, 6)( 3,11, 7)( 8,12)$
$3,2,1^{7}$ $24$ $6$ $( 1, 9, 5)( 8,12)$
$3^{2},2,1^{4}$ $48$ $6$ $( 1, 9, 5)( 3,11, 7)( 8,12)$
$3^{2},2,1^{4}$ $48$ $6$ $( 1, 9, 5)( 2,10, 6)( 8,12)$
$3^{3},2,1$ $96$ $6$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 8,12)$
$2^{2},1^{8}$ $36$ $2$ $( 7,11)( 8,12)$
$3,2^{2},1^{5}$ $72$ $6$ $( 2,10, 6)( 7,11)( 8,12)$
$3,2^{2},1^{5}$ $72$ $6$ $( 1, 9, 5)( 7,11)( 8,12)$
$3^{2},2^{2},1^{2}$ $144$ $6$ $( 1, 9, 5)( 2,10, 6)( 7,11)( 8,12)$
$2^{2},1^{8}$ $18$ $2$ $( 6,10)( 8,12)$
$3,2^{2},1^{5}$ $72$ $6$ $( 3,11, 7)( 6,10)( 8,12)$
$3^{2},2^{2},1^{2}$ $72$ $6$ $( 1, 9, 5)( 3,11, 7)( 6,10)( 8,12)$
$2^{3},1^{6}$ $108$ $2$ $( 6,10)( 7,11)( 8,12)$
$3,2^{3},1^{3}$ $216$ $6$ $( 1, 9, 5)( 6,10)( 7,11)( 8,12)$
$2^{4},1^{4}$ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$2^{6}$ $36$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$6,2^{3}$ $144$ $6$ $( 1, 7)( 2, 4,10,12, 6, 8)( 3, 9)( 5,11)$
$6^{2}$ $144$ $6$ $( 1, 3, 9,11, 5, 7)( 2, 4,10,12, 6, 8)$
$4,2^{4}$ $216$ $4$ $( 1, 7)( 2,12, 6, 8)( 3, 9)( 4,10)( 5,11)$
$6,4,2$ $432$ $12$ $( 1, 3, 9,11, 5, 7)( 2,12, 6, 8)( 4,10)$
$4^{2},2^{2}$ $324$ $4$ $( 1,11, 5, 7)( 2,12, 6, 8)( 3, 9)( 4,10)$
$4^{3}$ $216$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$
$12$ $432$ $12$ $( 1,12, 3, 6, 9, 8,11, 2, 5, 4, 7,10)$
$8,4$ $648$ $8$ $( 1, 4, 7,10)( 2, 5,12, 3, 6, 9, 8,11)$
$4^{3}$ $216$ $4$ $( 1,10, 7, 4)( 2,11, 8, 5)( 3,12, 9, 6)$
$12$ $432$ $12$ $( 1,10, 7,12, 9, 6, 3, 8, 5, 2,11, 4)$
$8,4$ $648$ $8$ $( 1,10, 7, 4)( 2,11,12, 9, 6, 3, 8, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5184=2^{6} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5184.bw
magma: IdentifyGroup(G);
 
Character table:    36 x 36 character table

magma: CharacterTable(G);