Properties

Label 8T22
Order \(32\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3 : D_4 $

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $22$
Group :  $C_2^3 : D_4 $
CHM label :  $E(8):D_{4}=[2^{3}]2^{2}$
Parity:  $1$
Primitive:  No
Generators:   (2,3)(6,7), (2,3)(4,5), (1,3)(2,8)(4,6)(5,7), (1,5)(2,6)(3,7)(4,8), (1,8)(2,3)(4,5)(6,7)
$|\textrm{Aut}(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3
16: 16T3

Subfields

Degree 2: $C_2$, $C_2$, $C_2$

Degree 4: $V_4$

Low degree siblings

8T22b, 8T22c, 8T22d, 8T22e, 8T22f, 16T23a, 16T23b, 16T23c, 16T23d, 16T23e, 16T23f, 16T23g, 16T23h, 16T23i
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 $ $2$ $2$ $(4,5)(6,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,3)(6,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,3)(4,5)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $2$ $4$ $(1,2,8,3)(4,6,5,7)$
$ 4, 4 $ $2$ $4$ $(1,2,8,3)(4,7,5,6)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,4)(2,6)(3,7)(5,8)$
$ 4, 4 $ $2$ $4$ $(1,4,8,5)(2,6,3,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $2$ $4$ $(1,4,8,5)(2,7,3,6)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,6)(2,4)(3,5)(7,8)$
$ 4, 4 $ $2$ $4$ $(1,6,8,7)(2,4,3,5)$
$ 4, 4 $ $2$ $4$ $(1,6,8,7)(2,5,3,4)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,6)(2,5)(3,4)(7,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,8)(2,3)(4,5)(6,7)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
Character table:  
      2  5  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  5

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

X.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
X.3      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
X.5      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
X.7      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
X.9      1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
X.10     1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1
X.11     1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1
X.12     1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1
X.13     1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
X.14     1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1
X.15     1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1
X.16     1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1
X.17     4  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . -4