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Properties

Label 8T17
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 17);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $17$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\wr C_2$
CHM label:  $[4^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $3$
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,8), (1,5)(2,6)(3,7)(4,8)
magma: Generators(G);
 

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: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T17, 16T28, 16T42, 32T14

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  32.11
magma: IdentifyGroup(G);
 
Character table:    
      2  5   4  4   4  5   4  4  5   4  3  3  3  3  5

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

X.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
X.3      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
X.5      1   A -1  -A -1  -A  1  1   A -1 -A  1  A -1
X.6      1  -A -1   A -1   A  1  1  -A -1  A  1 -A -1
X.7      1   A -1  -A -1  -A  1  1   A  1  A -1 -A -1
X.8      1  -A -1   A -1   A  1  1  -A  1 -A -1  A -1
X.9      2   . -2   .  2   . -2  2   .  .  .  .  .  2
X.10     2   .  2   . -2   . -2  2   .  .  .  .  . -2
X.11     2   B  .  /B  C -/B  . -2  -B  .  .  .  . -C
X.12     2  /B  .   B -C  -B  . -2 -/B  .  .  .  .  C
X.13     2 -/B  .  -B -C   B  . -2  /B  .  .  .  .  C
X.14     2  -B  . -/B  C  /B  . -2   B  .  .  .  . -C

A = -E(4)
  = -Sqrt(-1) = -i
B = -1+E(4)
  = -1+Sqrt(-1) = -1+i
C = -2*E(4)
  = -2*Sqrt(-1) = -2i

magma: CharacterTable(G);