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 | |
| Nilpotency class: | $1$ | |
| Generators: | (1,2,3,4,5,6) | |
| $|\Aut(F/K)|$: | $6$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 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
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