Properties

Label 24T21
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\SL(2,3):C_2$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $21$
Group :  $\SL(2,3):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,23)(2,4,24)(5,14,22)(6,13,21)(7,10,12)(8,9,11)(15,18,19)(16,17,20), (1,12,10,13,18,15,2,11,9,14,17,16)(3,20,5,21,7,23,4,19,6,22,8,24)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 8: None

Degree 12: $A_4\times C_2$

Low degree siblings

16T60

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 33]
Character table:   
      2  4  3  4   2   2   2   2   2   2   2   2  3  4  4
      3  1  .  1   1   1   1   1   1   1   1   1  .  1  1

        1a 2a 2b 12a  3a 12b  6a 12c  3b 12d  6b 4a 4b 4c
     2P 1a 1a 1a  6b  3b  6b  3b  6a  3a  6a  3a 2b 2b 2b
     3P 1a 2a 2b  4b  1a  4c  2b  4b  1a  4c  2b 4a 4c 4b
     5P 1a 2a 2b 12c  3b 12d  6b 12a  3a 12b  6a 4a 4b 4c
     7P 1a 2a 2b 12b  3a 12a  6a 12d  3b 12c  6b 4a 4c 4b
    11P 1a 2a 2b 12d  3b 12c  6b 12b  3a 12a  6a 4a 4c 4b

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   A  -A   A  -A  /A -/A  /A -/A  1 -1 -1
X.4      1 -1  1  /A -/A  /A -/A   A  -A   A  -A  1 -1 -1
X.5      1  1  1 -/A -/A -/A -/A  -A  -A  -A  -A  1  1  1
X.6      1  1  1  -A  -A  -A  -A -/A -/A -/A -/A  1  1  1
X.7      2  . -2   B  -1  -B   1   B  -1  -B   1  .  D -D
X.8      2  . -2  -B  -1   B   1  -B  -1   B   1  . -D  D
X.9      2  . -2   C   A  -C  -A -/C  /A  /C -/A  .  D -D
X.10     2  . -2 -/C  /A  /C -/A   C   A  -C  -A  .  D -D
X.11     2  . -2  /C  /A -/C -/A  -C   A   C  -A  . -D  D
X.12     2  . -2  -C   A   C  -A  /C  /A -/C -/A  . -D  D
X.13     3 -1  3   .   .   .   .   .   .   .   . -1  3  3
X.14     3  1  3   .   .   .   .   .   .   .   . -1 -3 -3

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(12)^7
D = -2*E(4)
  = -2*Sqrt(-1) = -2i