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Properties

Label 8T15
Order \(32\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Z_8 : Z_8^\times$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $15$
Group :  $Z_8 : Z_8^\times$
CHM label :  $[1/4.cD(4)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2,3,4,5,6,7,8), (1,5)(3,7), (1,6)(2,5)(3,4)(7,8)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T15, 16T35, 16T38 x 2, 16T45, 32T21

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 43]
Character table:    
      2  5  3  4  3  3  3  3  3  4  4  5

        1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e
     2P 1a 1a 1a 1a 1a 4b 2e 4b 2e 2e 1a
     3P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e
     5P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e
     7P 1a 2a 2b 2c 2d 8a 4a 8b 4b 4c 2e

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1 -1 -1  1  1  1 -1  1
X.3      1 -1 -1  1  1  1 -1 -1  1 -1  1
X.4      1 -1  1 -1 -1  1 -1  1  1  1  1
X.5      1 -1  1 -1  1 -1  1 -1  1  1  1
X.6      1  1 -1 -1 -1  1  1 -1  1 -1  1
X.7      1  1 -1 -1  1 -1 -1  1  1 -1  1
X.8      1  1  1  1 -1 -1 -1 -1  1  1  1
X.9      2  .  2  .  .  .  .  . -2 -2  2
X.10     2  . -2  .  .  .  .  . -2  2  2
X.11     4  .  .  .  .  .  .  .  .  . -4