Properties

Label 6T1
Order \(6\)
n \(6\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_6$

Related objects

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Group action invariants

Degree $n$ :  $6$
Transitive number $t$ :  $1$
Group :  $C_6$
CHM label :  $C(6) = 6 = 3[x]2$
Parity:  $-1$
Primitive:  No
Generators:  (1,2,3,4,5,6)
$|\Aut(F/K)|$:  $6$
Low degree resolvents:  
2: $C_2$
3: $C_3$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Low degree siblings

There is no other low degree representation.
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 6 $ $1$ $6$ $(1,2,3,4,5,6)$
$ 3, 3 $ $1$ $3$ $(1,3,5)(2,4,6)$
$ 2, 2, 2 $ $1$ $2$ $(1,4)(2,5)(3,6)$
$ 3, 3 $ $1$ $3$ $(1,5,3)(2,6,4)$
$ 6 $ $1$ $6$ $(1,6,5,4,3,2)$

Group invariants

Order:  $6=2 \cdot 3$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [6, 2]
Character table:  
     2  1   1  1  1  1   1
     3  1   1  1  1  1   1

       1a  6a 3a 2a 3b  6b

X.1     1   1  1  1  1   1
X.2     1  -1  1 -1  1  -1
X.3     1   A /A  1  A  /A
X.4     1  -A /A -1  A -/A
X.5     1  /A  A  1 /A   A
X.6     1 -/A  A -1 /A  -A

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3