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Properties

Label 24T50
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times A_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $50$
Group :  $C_2^2\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,18,13,10,6)(2,21,17,14,9,5)(3,23,20,15,11,8)(4,24,19,16,12,7), (1,4,10,23,5,8)(2,3,9,24,6,7)(11,17,20,13,16,22)(12,18,19,14,15,21)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3, $A_4$, $A_4\times C_2$ x 3

Degree 8: None

Degree 12: $C_6\times C_2$, $A_4 \times C_2$ x 3, $C_2^2 \times A_4$ x 3

Low degree siblings

12T25 x 3, 12T26 x 2, 16T58, 24T49 x 3

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 49]
Character table:    
      2  4  4  4   2   2   2   2   2   2   2   2  4  4  4  4  4
      3  1  .  .   1   1   1   1   1   1   1   1  .  1  .  1  1

        1a 2a 2b  6a  6b  3a  6c  6d  6e  6f  3b 2c 2d 2e 2f 2g
     2P 1a 1a 1a  3b  3b  3b  3b  3a  3a  3a  3a 1a 1a 1a 1a 1a
     3P 1a 2a 2b  2d  2g  1a  2f  2d  2g  2f  1a 2c 2d 2e 2f 2g
     5P 1a 2a 2b  6d  6e  3b  6f  6a  6b  6c  3a 2c 2d 2e 2f 2g

X.1      1  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 -1  1
X.3      1  1 -1   1  -1   1  -1   1  -1  -1   1  1  1 -1 -1 -1
X.4      1  1  1  -1  -1   1   1  -1  -1   1   1 -1 -1 -1  1 -1
X.5      1  1 -1   A  -A  -A   A  /A -/A  /A -/A -1 -1  1 -1  1
X.6      1  1 -1  /A -/A -/A  /A   A  -A   A  -A -1 -1  1 -1  1
X.7      1  1 -1 -/A  /A -/A  /A  -A   A   A  -A  1  1 -1 -1 -1
X.8      1  1 -1  -A   A  -A   A -/A  /A  /A -/A  1  1 -1 -1 -1
X.9      1  1  1   A   A  -A  -A  /A  /A -/A -/A -1 -1 -1  1 -1
X.10     1  1  1  /A  /A -/A -/A   A   A  -A  -A -1 -1 -1  1 -1
X.11     1  1  1 -/A -/A -/A -/A  -A  -A  -A  -A  1  1  1  1  1
X.12     1  1  1  -A  -A  -A  -A -/A -/A -/A -/A  1  1  1  1  1
X.13     3 -1 -1   .   .   .   .   .   .   .   . -1  3 -1  3  3
X.14     3 -1 -1   .   .   .   .   .   .   .   .  1 -3  1  3 -3
X.15     3 -1  1   .   .   .   .   .   .   .   . -1  3  1 -3 -3
X.16     3 -1  1   .   .   .   .   .   .   .   .  1 -3 -1 -3  3

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