Properties

Label 8T31
Order \(64\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(((C_4 \times C_2): C_2):C_2):C_2$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $31$
Group :  $(((C_4 \times C_2): C_2):C_2):C_2$
CHM label :  $[2^{4}]E(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (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: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T29 x 6, 8T31, 16T127, 16T128 x 3, 16T129 x 3, 16T147, 16T149 x 6, 16T150 x 3, 32T136 x 3, 32T137 x 2, 32T163 x 3

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 $ $2$ $2$ $(3,7)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(3,7)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 4 $ $4$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 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 $ $4$ $2$ $(1,4)(2,3)(5,8)(6,7)$
$ 4, 2, 2 $ $8$ $4$ $(1,4,5,8)(2,3)(6,7)$
$ 4, 4 $ $4$ $4$ $(1,4,5,8)(2,3,6,7)$
$ 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, 138]
Character table:   
      2  6  4  5  5  5  4  4  3  4  4  3  4  4  3  4  6

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

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