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Properties

Label 12T128
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, 128);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $128$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\wr C_2$
CHM label:   $[1/4E(4)^{3}:3]S(3)$
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:  (1,10)(3,12)(4,7)(6,9), (1,7,10)(2,5,11)(3,6,9), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
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: None

Degree 3: $S_3$

Degree 4: None

Degree 6: None

Low degree siblings

8T42, 12T126, 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 $ $1$ $1$ $()$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 4, 7,10)( 5,11, 8)( 6, 9,12)$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 4,10, 7)( 5, 8,11)( 6,12, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $12$ $2$ $( 2, 3)( 5,12)( 6,11)( 8, 9)$
$ 6, 3, 2, 1 $ $48$ $6$ $( 2, 3)( 4, 7,10)( 5, 6, 8,12,11, 9)$
$ 6, 3, 2, 1 $ $48$ $6$ $( 2, 3)( 4,10, 7)( 5, 9,11,12, 8, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 2, 5)( 3,12)( 6, 9)( 8,11)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4, 5, 9)( 6,10, 8)( 7,11,12)$
$ 3, 3, 3, 3 $ $8$ $3$ $( 1, 2, 3)( 4, 8,12)( 5, 6, 7)( 9,10,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 2, 3)( 4,11, 6)( 5,12,10)( 7, 8, 9)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1, 2, 4,11)( 3, 6)( 5, 7, 8,10)( 9,12)$
$ 6, 6 $ $24$ $6$ $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$
$ 6, 6 $ $24$ $6$ $( 1, 2, 6,10,11,12)( 3, 4, 8, 9, 7, 5)$
$ 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$

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);