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Properties

Label 12T159
Degree $12$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 159);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $159$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4^2:C_4$
CHM label:  $[2^{5}]F_{18}(6)_{2}$
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), (1,5,9)(4,8,12), (2,6,10)(3,7,11), (1,2,12,3)(4,10,5,11)(6,8,7,9)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$36$:  $C_3\times (C_3 : C_4)$
$288$:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3\times C_3$

Low degree siblings

16T1027, 24T1487, 24T1488, 36T719, 36T724, 36T946, 36T947

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 $ $9$ $2$ $( 1,12)( 2, 3)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 5, 9)( 4, 8,12)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 4, 8)( 2, 3)( 5, 9,12)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 9, 5)( 4,12, 8)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 9, 4)( 2, 3)( 5,12, 8)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 8, 5,12, 9, 4)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 9)( 2,11, 6)( 3,10, 7)( 4, 8,12)$
$ 6, 6 $ $16$ $6$ $( 1, 5, 9,12, 4, 8)( 2,11, 6, 3,10, 7)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 9, 5)( 2,11, 6)( 3,10, 7)( 4,12, 8)$
$ 6, 6 $ $32$ $6$ $( 1, 9, 5,12, 8, 4)( 2,11, 6, 3,10, 7)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 9, 5)( 2, 6,11)( 3, 7,10)( 4,12, 8)$
$ 6, 6 $ $16$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 6,11, 3, 7,10)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 4, 4 $ $12$ $4$ $( 1, 3,12, 2)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $36$ $4$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 12 $ $48$ $12$ $( 1, 2, 4,10, 9, 6,12, 3, 5,11, 8, 7)$
$ 12 $ $48$ $12$ $( 1, 3, 5,11, 8, 7,12, 2, 4,10, 9, 6)$
$ 12 $ $48$ $12$ $( 1, 2, 8, 7, 5,11,12, 3, 9, 6, 4,10)$
$ 12 $ $48$ $12$ $( 1, 3, 9, 6, 4,10,12, 2, 8, 7, 5,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  576.8278
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);