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Properties

Label 24T20
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:OD_{16}$

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Group action invariants

Degree $n$:  $24$
Transitive number $t$:  $20$
Group:  $C_3:OD_{16}$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $12$
Generators:  (1,24,7,18,2,23,8,17)(3,22,10,15,4,21,9,16)(5,19,11,14,6,20,12,13), (1,12,9,7,5,3,2,11,10,8,6,4)(13,24,21,20,17,15,14,23,22,19,18,16)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_3 : C_4$ x 2
$16$:  $C_8:C_2$
$24$:  24T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 8: $C_8:C_2$

Degree 12: $C_3 : C_4$

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [48, 10]
Character table:    
      2  4  3  4   3   3   3   3  3  3  3  3  4  3  4  3  3  3  3
      3  1  1  1   1   1   1   1  1  1  1  1  1  1  1  .  .  .  .

        1a 2a 2b 12a 12b 12c 12d 3a 6a 6b 6c 4a 4b 4c 8a 8b 8c 8d
     2P 1a 1a 1a  6c  6c  6c  6c 3a 3a 3a 3a 2b 2b 2b 4a 4c 4c 4a
     3P 1a 2a 2b  4b  4a  4c  4b 1a 2a 2a 2b 4c 4b 4a 8c 8d 8a 8b
     5P 1a 2a 2b 12d 12b 12c 12a 3a 6b 6a 6c 4a 4b 4c 8a 8b 8c 8d
     7P 1a 2a 2b 12d 12c 12b 12a 3a 6a 6b 6c 4c 4b 4a 8c 8d 8a 8b
    11P 1a 2a 2b 12a 12c 12b 12d 3a 6b 6a 6c 4c 4b 4a 8c 8d 8a 8b

X.1      1  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 -1  1
X.3      1 -1  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 -1 -1
X.5      1 -1  1   1  -1  -1   1  1 -1 -1  1 -1  1 -1  C -C -C  C
X.6      1 -1  1   1  -1  -1   1  1 -1 -1  1 -1  1 -1 -C  C  C -C
X.7      1  1  1  -1  -1  -1  -1  1  1  1  1 -1 -1 -1  C  C -C -C
X.8      1  1  1  -1  -1  -1  -1  1  1  1  1 -1 -1 -1 -C -C  C  C
X.9      2 -2  2  -1   1   1  -1 -1  1  1 -1 -2  2 -2  .  .  .  .
X.10     2 -2  2   1  -1  -1   1 -1  1  1 -1  2 -2  2  .  .  .  .
X.11     2  2  2  -1  -1  -1  -1 -1 -1 -1 -1  2  2  2  .  .  .  .
X.12     2  2  2   1   1   1   1 -1 -1 -1 -1 -2 -2 -2  .  .  .  .
X.13     2  . -2   .   B  -B   .  2  .  . -2 -B  .  B  .  .  .  .
X.14     2  . -2   .  -B   B   .  2  .  . -2  B  . -B  .  .  .  .
X.15     2  . -2   A   C  -C  -A -1  D -D  1 -B  .  B  .  .  .  .
X.16     2  . -2   A  -C   C  -A -1 -D  D  1  B  . -B  .  .  .  .
X.17     2  . -2  -A   C  -C   A -1 -D  D  1 -B  .  B  .  .  .  .
X.18     2  . -2  -A  -C   C   A -1  D -D  1  B  . -B  .  .  .  .

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3
B = 2*E(4)
  = 2*Sqrt(-1) = 2i
C = -E(4)
  = -Sqrt(-1) = -i
D = E(3)-E(3)^2
  = Sqrt(-3) = i3