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Properties

Label 12T126
Degree $12$
Order $288$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(12, 126);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $126$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\wr C_2$
CHM label:   $[A_{4}^{2}]2=A_{4}wr2$
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:  (2,8)(4,10), (2,6,10)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

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

8T42, 12T128, 12T129, 16T708, 18T112, 18T113, 24T692, 24T694, 24T695, 24T702, 24T703, 24T704, 32T9306, 36T316, 36T318, 36T456, 36T457, 36T458, 36T459

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  288.1025
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 4A 6A1 6A-1 6B1 6B-1
Size 1 6 9 12 8 8 16 16 32 36 24 24 48 48
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 2B 3A1 3A-1 3B1 3B-1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 4A 2A 2A 2C 2C
Type
288.1025.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
288.1025.1b R 1 1 1 −1 1 1 1 1 1 −1 1 1 −1 −1
288.1025.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
288.1025.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
288.1025.1d1 C 1 1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3
288.1025.1d2 C 1 1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1
288.1025.2a R 2 2 2 0 −1 −1 2 2 −1 0 −1 −1 0 0
288.1025.2b1 C 2 2 2 0 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 −ζ3−1 −ζ3 0 0
288.1025.2b2 C 2 2 2 0 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 −ζ3 −ζ3−1 0 0
288.1025.6a R 6 2 −2 0 3 3 0 0 0 0 −1 −1 0 0
288.1025.6b1 C 6 2 −2 0 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1 0 0
288.1025.6b2 C 6 2 −2 0 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3 0 0
288.1025.9a R 9 −3 1 3 0 0 0 0 0 −1 0 0 0 0
288.1025.9b R 9 −3 1 −3 0 0 0 0 0 1 0 0 0 0

magma: CharacterTable(G);