Properties

Label 24T49
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 $t$ :  $49$
Group :  $C_2^2\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,21,23,17,20)(2,16,22,24,18,19)(3,9,11,5,8,13)(4,10,12,6,7,14), (1,19,18,12,9,3)(2,20,17,11,10,4)(5,23,22,16,14,8)(6,24,21,15,13,7)
$|\Aut(F/K)|$:  $8$

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: $C_2$

Degree 3: $C_3$

Degree 4: None

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

Degree 8: None

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

Low degree siblings

12T25 x 3, 12T26 x 2, 16T58, 24T49 x 2, 24T50

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

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  2e  2g  1a  2f  2g  2e  2f  1a 2c 2d 2e 2f 2g
     5P 1a 2a 2b  6e  6d  3b  6f  6b  6a  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 -1  3  3  3
X.14     3 -1 -1   .   .   .   .   .   .   .   .  1  1 -3  3 -3
X.15     3  1 -1   .   .   .   .   .   .   .   . -1  1  3 -3 -3
X.16     3  1 -1   .   .   .   .   .   .   .   .  1 -1 -3 -3  3

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