Properties

Label 8T13
Order \(24\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times C_2$

Related objects

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $13$
Group :  $A_4\times C_2$
CHM label :  $E(8):3=A(4)[x]2$
Parity:  $1$
Primitive:  No
Generators:   (1,2)(3,8)(4,7)(5,6), (1,6)(2,4,8,5,3,7)
$|\Aut(F/K)|$:  $2$
Low degree resolvents:  
2: 2T1
3: 3T1
6: 6T1
12: 4T4

Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Low degree siblings

6T6, 12T6, 12T7
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 13]
Character table:  
     2  3  1  1  3   1   1  3  3
     3  1  1  1  .   1   1  .  1

       1a 3a 3b 2a  6a  6b 2b 2c
    2P 1a 3b 3a 1a  3a  3b 1a 1a
    3P 1a 1a 1a 2a  2c  2c 2b 2c
    5P 1a 3b 3a 2a  6b  6a 2b 2c

X.1     1  1  1  1   1   1  1  1
X.2     1  1  1  1  -1  -1 -1 -1
X.3     1  A /A  1 -/A  -A -1 -1
X.4     1 /A  A  1  -A -/A -1 -1
X.5     1  A /A  1  /A   A  1  1
X.6     1 /A  A  1   A  /A  1  1
X.7     3  .  . -1   .   .  1 -3
X.8     3  .  . -1   .   . -1  3

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3