Properties

Label 8T1
Order \(8\)
n \(8\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_8$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $1$
Group :  $C_8$
CHM label :  $C(8)=8$
Parity:  $-1$
Primitive:  No
Generators:  (1,2,3,4,5,6,7,8)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 8 $ $1$ $8$ $(1,2,3,4,5,6,7,8)$
$ 4, 4 $ $1$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 8 $ $1$ $8$ $(1,4,7,2,5,8,3,6)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$
$ 8 $ $1$ $8$ $(1,6,3,8,5,2,7,4)$
$ 4, 4 $ $1$ $4$ $(1,7,5,3)(2,8,6,4)$
$ 8 $ $1$ $8$ $(1,8,7,6,5,4,3,2)$

Group invariants

Order:  $8=2^{3}$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [8, 1]
Character table:   
     2  3   3  3   3  3   3  3   3

       1a  8a 4a  8b 2a  8c 4b  8d

X.1     1   1  1   1  1   1  1   1
X.2     1  -1  1  -1  1  -1  1  -1
X.3     1   A -1  -A  1   A -1  -A
X.4     1  -A -1   A  1  -A -1   A
X.5     1   B  A -/B -1  -B -A  /B
X.6     1  -B  A  /B -1   B -A -/B
X.7     1 -/B -A   B -1  /B  A  -B
X.8     1  /B -A  -B -1 -/B  A   B

A = E(4)
  = Sqrt(-1) = i
B = E(8)