Properties

Label 8T27
Order \(64\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $27$
CHM label :  $[2^{4}]4$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,8)(4,5,6,7), (4,8)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
32: 8T19

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

8T27b, 8T28a, 8T28b, 16T130, 16T157a, 16T157b, 16T158a, 16T158b, 16T159a, 16T159b, 16T166, 16T170, 16T171, 16T172
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, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $4$ $2$ $(3,7)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 4, 4 $ $8$ $4$ $(1,2,3,4)(5,6,7,8)$
$ 8 $ $8$ $8$ $(1,2,3,4,5,6,7,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,3)(2,4,6,8)(5,7)$
$ 4, 4 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 4, 4 $ $8$ $4$ $(1,4,3,2)(5,8,7,6)$
$ 8 $ $8$ $8$ $(1,4,7,6,5,8,3,2)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  6  4  4  5  4  3  3  4  3  4  3  3  6

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

X.1      1  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  1  1
X.3      1 -1  1  1 -1  1 -1  1 -1  1  1 -1  1
X.4      1  1  1  1  1 -1 -1  1  1  1 -1 -1  1
X.5      1 -1  1  1 -1  A -A -1  1 -1 -A  A  1
X.6      1 -1  1  1 -1 -A  A -1  1 -1  A -A  1
X.7      1  1  1  1  1  A  A -1 -1 -1 -A -A  1
X.8      1  1  1  1  1 -A -A -1 -1 -1  A  A  1
X.9      2  . -2  2  .  .  . -2  .  2  .  .  2
X.10     2  . -2  2  .  .  .  2  . -2  .  .  2
X.11     4  .  . -4  .  .  .  .  .  .  .  .  4
X.12     4 -2  .  .  2  .  .  .  .  .  .  . -4
X.13     4  2  .  . -2  .  .  .  .  .  .  . -4

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