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Properties

Label 8T41
Order \(192\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $V_4^2:(S_3\times C_2)$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $41$
Group :  $V_4^2:(S_3\times C_2)$
CHM label :  $E(8):S_{4}=[E(4)^{2}:S_{3}]2=E(4)^{2}:D_{12}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8), (1,3)(4,5,6,7)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
24:  $S_4$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Low degree siblings

8T41, 12T108 x 2, 12T109 x 2, 12T110 x 2, 12T111 x 2, 16T435 x 2, 16T436, 24T516 x 2, 24T517 x 2, 24T518 x 2, 24T519 x 2, 24T520 x 2, 24T521 x 2, 24T522 x 2, 24T523, 24T524 x 2, 24T525 x 2, 24T526 x 2, 24T527, 24T528, 24T529, 32T2148, 32T2149 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, 1, 1, 1, 1 $ $12$ $2$ $(3,8)(5,6)$
$ 4, 2, 1, 1 $ $24$ $4$ $(3,8)(4,5,7,6)$
$ 3, 3, 1, 1 $ $32$ $3$ $(2,3,8)(5,7,6)$
$ 2, 2, 2, 2 $ $6$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $3$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $12$ $4$ $(1,2,3,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $12$ $2$ $(1,4)(2,5)(3,6)(7,8)$
$ 4, 4 $ $24$ $4$ $(1,4,2,5)(3,6,8,7)$
$ 4, 4 $ $12$ $4$ $(1,4,3,6)(2,5,8,7)$
$ 6, 2 $ $32$ $6$ $(1,4)(2,5,3,7,8,6)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $12$ $4$ $(1,4,2,7)(3,6,8,5)$

Group invariants

Order:  $192=2^{6} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [192, 955]
Character table:    
      2  6  5  4  3  1  5  6  4  4  3  4  1  4  4
      3  1  .  .  .  1  .  .  .  .  .  .  1  1  .

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

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      2  2  .  . -1  2  2  .  .  .  . -1  2  2
X.6      2  2  .  . -1  2  2  .  .  .  .  1 -2 -2
X.7      3 -1 -1  1  . -1  3 -1 -1  1 -1  .  3 -1
X.8      3 -1 -1  1  . -1  3 -1  1 -1  1  . -3  1
X.9      3 -1  1 -1  . -1  3  1 -1  1 -1  . -3  1
X.10     3 -1  1 -1  . -1  3  1  1 -1  1  .  3 -1
X.11     6 -2  .  .  .  2 -2  . -2  .  2  .  .  .
X.12     6 -2  .  .  .  2 -2  .  2  . -2  .  .  .
X.13     6  2 -2  .  . -2 -2  2  .  .  .  .  .  .
X.14     6  2  2  .  . -2 -2 -2  .  .  .  .  .  .