# Properties

 Label 12T2 Degree $12$ Order $12$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group no Group: $C_6\times C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(12, 2);

## Group action invariants

 Degree $n$: $12$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $2$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_6\times C_2$ CHM label: $E(4)[x]C(3)=6x2$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $12$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) 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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $6, 6$ $1$ $6$ $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ $6, 6$ $1$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ $3, 3, 3, 3$ $1$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ $6, 6$ $1$ $6$ $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ $6, 6$ $1$ $6$ $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ $3, 3, 3, 3$ $1$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$ $6, 6$ $1$ $6$ $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ $6, 6$ $1$ $6$ $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $12=2^{2} \cdot 3$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 12.5 magma: IdentifyGroup(G);
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1a 6a 6b 2a 3a 6c 2b 6d 3b 2c 6e 6f 2P 1a 3b 3a 1a 3b 3a 1a 3b 3a 1a 3b 3a 3P 1a 2c 2b 2a 1a 2c 2b 2a 1a 2c 2b 2a 5P 1a 6c 6e 2a 3b 6a 2b 6f 3a 2c 6b 6d X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.5 1 A /A 1 -A /A -1 -A -/A -1 A -/A X.6 1 /A A 1 -/A A -1 -/A -A -1 /A -A X.7 1 A -/A -1 -A /A 1 A -/A -1 -A /A X.8 1 /A -A -1 -/A A 1 /A -A -1 -/A A X.9 1 -/A A -1 -/A -A -1 /A -A 1 /A A X.10 1 -A /A -1 -A -/A -1 A -/A 1 A /A X.11 1 -/A -A 1 -/A -A 1 -/A -A 1 -/A -A X.12 1 -A -/A 1 -A -/A 1 -A -/A 1 -A -/A A = -E(3) = (1-Sqrt(-3))/2 = -b3 

magma: CharacterTable(G);