↑

Properties

Label 8T26
Order \(64\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_4^2 : C_2):C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $26$
Group :  $(C_4^2 : C_2):C_2$
CHM label :  $1/2[2^{4}]eD(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2,3,4,5,6,7,8), (1,7)(3,5)(4,8), (1,5)(4,8)
$|\Aut(F/K)|$:  $2$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T26 x 3, 16T135 x 2, 16T141 x 2, 16T142 x 2, 16T152 x 2, 32T147 x 2, 32T148 x 2, 32T155, 32T156

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, 2, 1, 1, 1, 1 $ $4$ $2$ $(3,7)(4,8)$
$ 4, 1, 1, 1, 1 $ $4$ $4$ $(2,4,6,8)$
$ 2, 2, 2, 1, 1 $ $8$ $2$ $(2,4)(3,7)(6,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 8 $ $8$ $8$ $(1,2,3,4,5,6,7,8)$
$ 8 $ $8$ $8$ $(1,2,3,8,5,6,7,4)$
$ 4, 4 $ $4$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 4, 4 $ $4$ $4$ $(1,2,5,6)(3,8,7,4)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 4 $ $2$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 4, 2, 2 $ $4$ $4$ $(1,3,5,7)(2,6)(4,8)$
$ 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:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 134]
Character table:    
      2  6  4  4  3  5  4  4  3  3  4  4  4  5  4  5  6

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2 -2  .  .  2  .  .  .  .  .  .  2 -2  . -2  2
X.10     2  2  .  .  2  .  .  .  .  .  . -2 -2  . -2  2
X.11     2  .  .  . -2 -2  .  .  .  2  .  . -2  .  2  2
X.12     2  .  .  . -2  . -2  .  .  .  2  .  2  . -2  2
X.13     2  .  .  . -2  .  2  .  .  . -2  .  2  . -2  2
X.14     2  .  .  . -2  2  .  .  . -2  .  . -2  .  2  2
X.15     4  . -2  .  .  .  .  .  .  .  .  .  .  2  . -4
X.16     4  .  2  .  .  .  .  .  .  .  .  .  . -2  . -4