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Properties

Label 12T29
Degree $12$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_4\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(12, 29);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\times A_4$
CHM label:   $[1/2.4^{2}]3$
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:  (1,7)(3,9)(4,10)(6,12), (1,5,9)(2,6,10)(3,7,11)(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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $A_4$, $C_{12}$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

16T57, 24T55, 24T56

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

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:  48.31
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A1 4A-1 4B1 4B-1 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 3 3 4 4 1 1 3 3 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A-1 3A1 2A 2A 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1
3 P 1A 2A 2B 2C 1A 1A 4A-1 4A1 4B-1 4B1 2A 2A 4A1 4A-1 4A1 4A-1
Type
48.31.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.31.1b R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1
48.31.1c1 C 1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
48.31.1c2 C 1 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
48.31.1d1 C 1 −1 −1 1 1 1 −i i −i i −1 −1 i −i i −i
48.31.1d2 C 1 −1 −1 1 1 1 i −i i −i −1 −1 −i i −i i
48.31.1e1 C 1 1 1 1 ζ3−1 ζ3 −1 −1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.31.1e2 C 1 1 1 1 ζ3 ζ3−1 −1 −1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.31.1f1 C 1 −1 −1 1 −ζ122 ζ124 −ζ123 ζ123 −ζ123 ζ123 −ζ124 ζ122 −ζ125 ζ12 −ζ12 ζ125
48.31.1f2 C 1 −1 −1 1 ζ124 −ζ122 ζ123 −ζ123 ζ123 −ζ123 ζ122 −ζ124 ζ12 −ζ125 ζ125 −ζ12
48.31.1f3 C 1 −1 −1 1 −ζ122 ζ124 ζ123 −ζ123 ζ123 −ζ123 −ζ124 ζ122 ζ125 −ζ12 ζ12 −ζ125
48.31.1f4 C 1 −1 −1 1 ζ124 −ζ122 −ζ123 ζ123 −ζ123 ζ123 ζ122 −ζ124 −ζ12 ζ125 −ζ125 ζ12
48.31.3a R 3 3 −1 −1 0 0 3 3 −1 −1 0 0 0 0 0 0
48.31.3b R 3 3 −1 −1 0 0 −3 −3 1 1 0 0 0 0 0 0
48.31.3c1 C 3 −3 1 −1 0 0 −3i 3i i −i 0 0 0 0 0 0
48.31.3c2 C 3 −3 1 −1 0 0 3i −3i −i i 0 0 0 0 0 0

magma: CharacterTable(G);