Properties

Label 24T4
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times Q_8$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $4$
Group :  $C_3\times Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,4,18,19,9,12,2,3,17,20,10,11)(5,8,21,24,14,16,6,7,22,23,13,15), (1,6,10,14,17,21,2,5,9,13,18,22)(3,7,12,16,19,24,4,8,11,15,20,23)
$|\Aut(F/K)|$:  $24$

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
8:  $Q_8$
12:  $C_6\times C_2$

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

Degree 8: $Q_8$

Degree 12: $C_6\times C_2$

Low degree siblings

There are no siblings 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 12, 12 $ $2$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
$ 12, 12 $ $2$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 12, 12 $ $2$ $12$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,15,13,23,22, 7, 6,16,14,24,21, 8)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 2,14)( 3,15, 4,16)( 5,17, 6,18)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$
$ 12, 12 $ $2$ $12$ $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 5,20,21,11,14, 4, 6,19,22,12,13)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15) ( 8,23,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,19, 2,20)( 3,10, 4, 9)( 5,24, 6,23)( 7,13, 8,14)(11,18,12,17)(15,21,16,22)$
$ 12, 12 $ $2$ $12$ $( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,24,20,16,11, 7, 4,23,19,15,12, 8)$
$ 12, 12 $ $2$ $12$ $( 1,23,10, 7,17,16, 2,24, 9, 8,18,15)( 3,13,12,22,19, 6, 4,14,11,21,20, 5)$

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 11]
Character table:   
      2  3  3   2   2  2   3   3   2  2   2   3   3  2   2   2
      3  1  1   1   1  1   1   1   1  1   1   1   1  1   1   1

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

X.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
X.3      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
X.5      1  1   A  /A  1  -A  -A  /A -1  -A -/A -/A -1   A -/A
X.6      1  1  /A   A  1 -/A -/A   A -1 -/A  -A  -A -1  /A  -A
X.7      1  1   A -/A -1  -A  -A  /A  1   A -/A -/A -1  -A  /A
X.8      1  1  /A  -A -1 -/A -/A   A  1  /A  -A  -A -1 -/A   A
X.9      1  1 -/A   A -1 -/A -/A  -A -1  /A  -A  -A  1  /A   A
X.10     1  1  -A  /A -1  -A  -A -/A -1   A -/A -/A  1   A  /A
X.11     1  1 -/A  -A  1 -/A -/A  -A  1 -/A  -A  -A  1 -/A  -A
X.12     1  1  -A -/A  1  -A  -A -/A  1  -A -/A -/A  1  -A -/A
X.13     2 -2   .   .  .   2  -2   .  .   .   2  -2  .   .   .
X.14     2 -2   .   .  .   B  -B   .  .   .  /B -/B  .   .   .
X.15     2 -2   .   .  .  /B -/B   .  .   .   B  -B  .   .   .

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3