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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

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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
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)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Low degree siblings

6T6, 12T6, 12T7, 24T9

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 $ $1$ $1$ $()$
$ 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