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Properties

Label 8T21
Order \(32\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3: C_4$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $21$
Group :  $C_2^3: C_4$
CHM label :  $1/2[2^{4}]E(4)=[1/4.dD(4)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,4,5,8)(2,3)(6,7), (1,3)(2,8)(4,6)(5,7), (1,5)(3,7)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T19 x 2, 8T20, 16T33 x 2, 16T52, 16T53, 32T19

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 $ $2$ $2$ $(3,7)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(3,7)$
$ 4, 2, 2 $ $4$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 2, 2 $ $4$ $4$ $(1,2)(3,8,7,4)(5,6)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 4 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 4, 2, 2 $ $4$ $4$ $(1,4,5,8)(2,3)(6,7)$
$ 4, 2, 2 $ $4$ $4$ $(1,4)(2,3,6,7)(5,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 6]
Character table:    
      2  5  4  4  4  3  3  3  3  3  3  5

        1a 2a 2b 2c 4a 4b 2d 4c 4d 4e 2e
     2P 1a 1a 1a 1a 2a 2a 1a 2e 2c 2c 1a
     3P 1a 2a 2b 2c 4b 4a 2d 4c 4e 4d 2e

X.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
X.3      1  1  1  1 -1 -1  1  1 -1 -1  1
X.4      1  1  1  1  1  1 -1 -1 -1 -1  1
X.5      1 -1  1 -1  A -A -1  1 -A  A  1
X.6      1 -1  1 -1 -A  A -1  1  A -A  1
X.7      1 -1  1 -1  A -A  1 -1  A -A  1
X.8      1 -1  1 -1 -A  A  1 -1 -A  A  1
X.9      2  2 -2 -2  .  .  .  .  .  .  2
X.10     2 -2 -2  2  .  .  .  .  .  .  2
X.11     4  .  .  .  .  .  .  .  .  . -4

A = -E(4)
  = -Sqrt(-1) = -i