Properties

Label 8T3
Order \(8\)
n \(8\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $3$
Group :  $C_2^3$
CHM label :  $E(8)=2[x]2[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

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

Group invariants

Order:  $8=2^{3}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [8, 5]
Character table:   
     2  3  3  3  3  3  3  3  3

       1a 2a 2b 2c 2d 2e 2f 2g
    2P 1a 1a 1a 1a 1a 1a 1a 1a

X.1     1  1  1  1  1  1  1  1
X.2     1 -1 -1 -1 -1  1  1  1
X.3     1 -1 -1  1  1 -1 -1  1
X.4     1 -1  1 -1  1 -1  1 -1
X.5     1 -1  1  1 -1  1 -1 -1
X.6     1  1 -1 -1  1  1 -1 -1
X.7     1  1 -1  1 -1 -1  1 -1
X.8     1  1  1 -1 -1 -1 -1  1