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Properties

Label 8T46
Order \(576\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4^2:C_4$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $46$
Group :  $A_4^2:C_4$
CHM label :  $1/2[S(4)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(4,5), (1,5)(2,7,3,6)(4,8), (1,3)(2,8), (1,2,3)
$|\Aut(F/K)|$:  $1$

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 $ $72$ $8$ $(1,4,3,6,2,7,8,5)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8652]
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