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Properties

Label 8T46
Degree $8$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(8, 46);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4^2:C_4$
CHM label:  $1/2[S(4)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(4,5), (1,5)(2,7,3,6)(4,8), (1,3)(2,8), (1,2,3)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T160, 12T162, 16T1030, 16T1031, 18T182, 18T184, 24T1489, 24T1491, 24T1505, 24T1506 x 2, 24T1508, 32T34594, 36T764, 36T765, 36T766, 36T767, 36T964, 36T965

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 $ $6$ $2$ $(4,5)(6,7)$
$ 2, 2, 2, 2 $ $9$ $2$ $(1,8)(2,3)(4,5)(6,7)$
$ 3, 1, 1, 1, 1, 1 $ $16$ $3$ $(2,3,8)$
$ 3, 2, 2, 1 $ $48$ $6$ $(2,3,8)(4,5)(6,7)$
$ 3, 3, 1, 1 $ $64$ $3$ $(2,3,8)(5,6,7)$
$ 2, 2, 1, 1, 1, 1 $ $36$ $2$ $(3,8)(6,7)$
$ 4, 2, 1, 1 $ $72$ $4$ $(1,8,2,3)(6,7)$
$ 4, 4 $ $36$ $4$ $(1,8,2,3)(4,7,5,6)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,7,3,6)(4,8)$
$ 8 $ $72$ $8$ $(1,5,3,6,8,4,2,7)$
$ 4, 2, 2 $ $72$ $4$ $(1,5)(2,6)(3,7,8,4)$
$ 8 $ $72$ $8$ $(1,4,3,6,2,7,8,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  576.8652
magma: IdentifyGroup(G);
 
Character table:    
      2  6  5  6  2  2  .  4  3  4  3  3  3  3
      3  2  1  .  2  1  2  .  .  .  .  .  .  .

        1a 2a 2b 3a 6a 3b 2c 4a 4b 4c 8a 4d 8b
     2P 1a 1a 1a 3a 3a 3b 1a 2a 2b 2c 4b 2c 4b
     3P 1a 2a 2b 1a 2a 1a 2c 4a 4b 4d 8b 4c 8a
     5P 1a 2a 2b 3a 6a 3b 2c 4a 4b 4c 8a 4d 8b
     7P 1a 2a 2b 3a 6a 3b 2c 4a 4b 4d 8b 4c 8a

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

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

magma: CharacterTable(G);