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Properties

Label 8T12
Order \(24\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\SL(2,3)$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $12$
Group :  $\SL(2,3)$
CHM label :  $2A_{4}(8)=[2]A(4)=SL(2,3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5,7)(2,4,6,8), (1,3,8)(4,5,7)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Low degree siblings

24T7

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

Group invariants

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

       1a  3a  3b 4a 6a 6b 2a
    2P 1a  3b  3a 2a 3a 3b 1a
    3P 1a  1a  1a 4a 2a 2a 2a
    5P 1a  3b  3a 4a 6b 6a 2a

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

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