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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

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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
Nilpotency class:  $4$
Generators:  (1,2,3,8)(4,5,6,7), (4,8), (1,3)(5,7)
$|\Aut(F/K)|$:  $2$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T35 x 7, 16T376 x 4, 16T388 x 4, 16T390 x 4, 16T391 x 4, 16T393 x 4, 16T395 x 4, 16T396 x 4, 16T401 x 4, 32T852 x 4, 32T853 x 2, 32T854 x 2, 32T872 x 2, 32T876 x 4, 32T877 x 2, 32T880 x 2, 32T882 x 2, 32T883 x 4, 32T884 x 2, 32T885 x 2

Siblings are shown 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, 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
GAP id:  [128, 928]
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