Properties

Label 24T7
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\SL(2,3)$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $7$
Group :  $\SL(2,3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,16)(2,6,15)(3,19,11)(4,20,12)(7,18,22)(8,17,21)(9,14,24)(10,13,23), (1,19,2,20)(3,10,4,9)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,22,16,21)
$|\Aut(F/K)|$:  $24$

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 3: $C_3$

Degree 4: $A_4$

Degree 6: $A_4$

Degree 8: $\SL(2,3)$

Degree 12: $A_4$

Low degree siblings

8T12

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3,23)( 2, 4,24)( 5,14,22)( 6,13,21)( 7,10,12)( 8, 9,11)(15,18,19) (16,17,20)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1, 4,23, 2, 3,24)( 5,13,22, 6,14,21)( 7, 9,12, 8,10,11)(15,17,19,16,18,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 5,16)( 2, 6,15)( 3,19,11)( 4,20,12)( 7,18,22)( 8,17,21)( 9,14,24) (10,13,23)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1, 6,16, 2, 5,15)( 3,20,11, 4,19,12)( 7,17,22, 8,18,21)( 9,13,24,10,14,23)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,16,10,15)(13,19,14,20)(17,23,18,24)$

Group invariants

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

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

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

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