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Properties

Label 8T45
Order \(576\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(A_4\wr C_2):C_2$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $45$
Group :  $(A_4\wr C_2):C_2$
CHM label :  $[1/2.S(4)^{2}]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(4,5), (1,3)(2,8), (1,2,3), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

12T161, 12T163, 12T165 x 2, 16T1032, 16T1034, 18T179, 18T180, 18T185 x 2, 24T1490, 24T1492, 24T1493 x 2, 24T1494 x 2, 24T1495 x 2, 24T1503, 24T1504 x 2, 32T34597 x 2, 32T34598, 36T759, 36T760, 36T762, 36T763, 36T774 x 2, 36T775 x 2, 36T960, 36T961, 36T962 x 2, 36T963 x 2

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 $ $32$ $3$ $(2,3,8)(5,7,6)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,8,3)(5,7,6)$
$ 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 $ $36$ $4$ $(1,8,2,3)(4,6,5,7)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,5)(2,6)(3,7)(4,8)$
$ 4, 4 $ $36$ $4$ $(1,4,8,5)(2,7,3,6)$
$ 6, 2 $ $96$ $6$ $(1,5)(2,6,3,7,8,4)$
$ 6, 2 $ $96$ $6$ $(1,5)(2,7,8,4,3,6)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,5)(2,7)(3,6)(4,8)$
$ 4, 4 $ $36$ $4$ $(1,4,8,5)(2,6,3,7)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8654]
Character table:    
      2  6  5  6  2  2  1  1  4  3  4  4  4  1  1  4  4
      3  2  1  .  2  1  2  2  .  .  .  1  .  1  1  1  .

        1a 2a 2b 3a 6a 3b 3c 2c 4a 4b 2d 4c 6b 6c 2e 4d
     2P 1a 1a 1a 3a 3a 3b 3c 1a 2a 2b 1a 2b 3b 3c 1a 2b
     3P 1a 2a 2b 1a 2a 1a 1a 2c 4a 4b 2d 4c 2d 2e 2e 4d
     5P 1a 2a 2b 3a 6a 3b 3c 2c 4a 4b 2d 4c 6b 6c 2e 4d

X.1      1  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  1  1
X.3      1  1  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 -1 -1
X.5      2  2  2 -1 -1 -1  2  .  .  .  2  2 -1  .  .  .
X.6      2  2  2 -1 -1 -1  2  .  .  . -2 -2  1  .  .  .
X.7      2  2  2 -1 -1  2 -1  .  .  .  .  .  . -1  2  2
X.8      2  2  2 -1 -1  2 -1  .  .  .  .  .  .  1 -2 -2
X.9      4  4  4  1  1 -2 -2  .  .  .  .  .  .  .  .  .
X.10     6  2 -2  3 -1  .  . -2  .  2  .  .  .  .  .  .
X.11     6  2 -2  3 -1  .  .  2  . -2  .  .  .  .  .  .
X.12     9 -3  1  .  .  .  . -1  1 -1 -3  1  .  .  3 -1
X.13     9 -3  1  .  .  .  . -1  1 -1  3 -1  .  . -3  1
X.14     9 -3  1  .  .  .  .  1 -1  1 -3  1  .  . -3  1
X.15     9 -3  1  .  .  .  .  1 -1  1  3 -1  .  .  3 -1
X.16    12  4 -4 -3  1  .  .  .  .  .  .  .  .  .  .  .