Properties

Label 8T35
Order \(128\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2 \wr C_2\wr C_2$

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $35$
Group :  $C_2 \wr C_2\wr C_2$
CHM label :  $[2^{4}]D(4)$
Parity:  $-1$
Primitive:  No
Generators:   (1,2,3,8)(4,5,6,7), (4,8), (1,3)(5,7)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18
64: 8T29

Subfields

Degree 2: $C_2$

Degree 4: $D_4$

Low degree siblings

8T35b, 8T35c, 8T35d, 8T35e, 8T35f, 8T35g, 8T35h, 16T376a, 16T376b, 16T376c, 16T376d, 16T388a, 16T388b, 16T388c, 16T388d, 16T390a, 16T390b, 16T390c, 16T390d, 16T391a, 16T391b, 16T391c, 16T391d, 16T393a, 16T393b, 16T393c, 16T393d, 16T395a, 16T395b, 16T395c, 16T395d, 16T396a, 16T396b, 16T396c, 16T396d, 16T401a, 16T401b, 16T401c, 16T401d
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 $ $4$ $2$ $(2,4)(6,8)$
$ 4, 1, 1, 1, 1 $ $4$ $4$ $(2,4,6,8)$
$ 2, 2, 2, 1, 1 $ $8$ $2$ $(2,4)(3,7)(6,8)$
$ 4, 2, 1, 1 $ $8$ $4$ $(2,4,6,8)(3,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $8$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $16$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 4 $ $16$ $4$ $(1,2,3,4)(5,6,7,8)$
$ 8 $ $16$ $8$ $(1,2,3,4,5,6,7,8)$
$ 4, 4 $ $8$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 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 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,6)(4,8)(5,7)$
$ 4, 2, 2 $ $4$ $4$ $(1,3,5,7)(2,6)(4,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  7  5  5  5  5  4  4  6  5  4  3  3  3  4  5  4  5  5  5  7

        1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     2P 1a 1a 1a 1a 2e 1a 2e 1a 1a 1a 2b 2h 4g 2j 1a 2e 2j 1a 2e 1a
     3P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     5P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     7P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j

X.1      1  1  1  1  1  1  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 -1  1 -1  1 -1  1  1
X.3      1 -1  1 -1  1  1 -1  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 -1  1 -1  1  1 -1  1
X.5      1 -1  1  1 -1 -1  1  1 -1  1 -1  1 -1  1  1 -1  1  1 -1  1
X.6      1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1  1  1  1 -1 -1  1
X.7      1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1  1  1 -1 -1  1
X.8      1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1  1  1  1  1  1  1
X.9      2  2  2  .  .  .  .  2  2  .  .  .  .  . -2 -2 -2  .  .  2
X.10     2 -2  2  .  .  .  .  2 -2  .  .  .  .  . -2  2 -2  .  .  2
X.11     2  . -2  2  .  . -2  2  .  .  .  .  .  .  2  . -2  2  .  2
X.12     2  . -2 -2  .  .  2  2  .  .  .  .  .  .  2  . -2 -2  .  2
X.13     2  . -2  . -2  2  .  2  .  .  .  .  .  . -2  .  2  . -2  2
X.14     2  . -2  .  2 -2  .  2  .  .  .  .  .  . -2  .  2  .  2  2
X.15     4  .  .  .  .  .  . -4  . -2  .  .  .  2  .  .  .  .  .  4
X.16     4  .  .  .  .  .  . -4  .  2  .  .  . -2  .  .  .  .  .  4
X.17     4 -2  . -2  2  .  .  .  2  .  .  .  .  .  .  .  .  2 -2 -4
X.18     4 -2  .  2 -2  .  .  .  2  .  .  .  .  .  .  .  . -2  2 -4
X.19     4  2  . -2 -2  .  .  . -2  .  .  .  .  .  .  .  .  2  2 -4
X.20     4  2  .  2  2  .  .  . -2  .  .  .  .  .  .  .  . -2 -2 -4