# Properties

 Label 12T1 Order $$12$$ n $$12$$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group No Group: $C_{12}$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $1$ Group : $C_{12}$ CHM label : $C(4)[x]C(3)$ Parity: $-1$ Primitive: No Nilpotency class: $1$ Generators: (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) $|\Aut(F/K)|$: $12$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $C_4$

Degree 6: $C_6$

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

## Group invariants

 Order: $12=2^{2} \cdot 3$ Cyclic: Yes Abelian: Yes Solvable: Yes GAP id: [12, 2]
 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 12a 6a 4a 3a 12b 2a 12c 3b 4b 6b 12d 2P 1a 6a 3a 2a 3b 6b 1a 6a 3a 2a 3b 6b 3P 1a 4a 2a 4b 1a 4a 2a 4b 1a 4a 2a 4b 5P 1a 12b 6b 4a 3b 12a 2a 12d 3a 4b 6a 12c 7P 1a 12c 6a 4b 3a 12d 2a 12a 3b 4a 6b 12b 11P 1a 12d 6b 4b 3b 12c 2a 12b 3a 4a 6a 12a 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 A -/A -1 -A /A 1 A -/A -1 -A /A X.4 1 /A -A -1 -/A A 1 /A -A -1 -/A A 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 B -1 -B 1 B -1 -B 1 B -1 -B X.8 1 -B -1 B 1 -B -1 B 1 -B -1 B X.9 1 C /A -B -A -/C -1 -C -/A B A /C X.10 1 -/C A -B -/A C -1 /C -A B /A -C X.11 1 /C A B -/A -C -1 -/C -A -B /A C X.12 1 -C /A B -A /C -1 C -/A -B A -/C A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = -E(4) = -Sqrt(-1) = -i C = -E(12)^7