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Properties

Label 8T27
Degree $8$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $((C_8 : C_2):C_2):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 27);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $((C_8 : C_2):C_2):C_2$
CHM label:  $[2^{4}]4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,8)(4,5,6,7), (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, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

8T27, 8T28 x 2, 16T130, 16T157 x 2, 16T158 x 2, 16T159 x 2, 16T166, 16T170, 16T171, 16T172, 32T138 x 2, 32T139, 32T170, 32T176

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $64=2^{6}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $4$
Label:  64.32
magma: IdentifyGroup(G);
 
Character table:    
      2  6  4  4  5  4  3  3  4  3  4  3  3  6

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

X.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
X.3      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
X.5      1 -1  1  1 -1  A -A -1  1 -1 -A  A  1
X.6      1 -1  1  1 -1 -A  A -1  1 -1  A -A  1
X.7      1  1  1  1  1  A  A -1 -1 -1 -A -A  1
X.8      1  1  1  1  1 -A -A -1 -1 -1  A  A  1
X.9      2  . -2  2  .  .  . -2  .  2  .  .  2
X.10     2  . -2  2  .  .  .  2  . -2  .  .  2
X.11     4  .  . -4  .  .  .  .  .  .  .  .  4
X.12     4 -2  .  .  2  .  .  .  .  .  .  . -4
X.13     4  2  .  . -2  .  .  .  .  .  .  . -4

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

magma: CharacterTable(G);