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Properties

Label 8T2
Order \(8\)
n \(8\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $2$
Group :  $C_4\times C_2$
CHM label :  $4[x]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,2,3,8)(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 3
4:  $C_4$ x 2, $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

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

Group invariants

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

       1a 4a 2a 4b 2b 4c 2c 4d
    2P 1a 2a 1a 2a 1a 2a 1a 2a
    3P 1a 4d 2a 4c 2b 4b 2c 4a

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  A -1  A -1 -A  1 -A
X.6     1 -A -1 -A -1  A  1  A
X.7     1  A -1 -A  1  A -1 -A
X.8     1 -A -1  A  1 -A -1  A

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