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Properties

Label 12T26
Degree $12$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2 \times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 26);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2 \times A_4$
CHM label:   $A_{4}(12)x2^{2}$
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:  (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,4)(2,5)(3,9)(6,12)(7,10)(8,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4\times C_2$ x 3

Low degree siblings

12T25 x 3, 12T26, 16T58, 24T49 x 3, 24T50

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 5)( 3, 6)( 8,11)( 9,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3,12)( 5,11)( 6, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2,11)( 3, 9)( 5, 8)( 6,12)$
$ 6, 6 $ $4$ $6$ $( 1, 2, 3,10, 8, 6)( 4, 5, 9, 7,11,12)$
$ 6, 6 $ $4$ $6$ $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$
$ 6, 6 $ $4$ $6$ $( 1, 2, 9, 4, 5, 3)( 6,10, 8,12, 7,11)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$
$ 6, 6 $ $4$ $6$ $( 1, 3, 5, 4, 9, 2)( 6,11, 7,12, 8,10)$
$ 6, 6 $ $4$ $6$ $( 1, 3,11,10, 6, 5)( 2, 4, 9, 8, 7,12)$
$ 6, 6 $ $4$ $6$ $( 1, 3, 2, 7,12,11)( 4, 9, 5,10, 6, 8)$
$ 3, 3, 3, 3 $ $4$ $3$ $( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$
$ 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2,11)( 3,12)( 4,10)( 5, 8)( 6, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 8)( 3, 6)( 4, 7)( 5,11)( 9,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1
Size 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1
3 P 1A 2C 2A 2B 2G 2D 2F 2E 1A 1A 2B 2A 2B 2A 2C 2C
Type
48.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.49.1b R 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
48.49.1c R 1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
48.49.1d R 1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1
48.49.1e1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
48.49.1e2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
48.49.1f1 C 1 −1 −1 1 −1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3
48.49.1f2 C 1 −1 −1 1 −1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
48.49.1g1 C 1 −1 1 −1 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
48.49.1g2 C 1 −1 1 −1 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
48.49.1h1 C 1 1 −1 −1 −1 1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
48.49.1h2 C 1 1 −1 −1 −1 1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
48.49.3a R 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0
48.49.3b R 3 −3 −3 3 1 1 −1 −1 0 0 0 0 0 0 0 0
48.49.3c R 3 −3 3 −3 −1 1 1 −1 0 0 0 0 0 0 0 0
48.49.3d R 3 3 −3 −3 1 −1 1 −1 0 0 0 0 0 0 0 0

magma: CharacterTable(G);