Properties

Label 24T31
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{24}:C_2$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $31$
Group :  $C_{24}:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,22,19,18,16,13,12,9,7,5,3,2,23,21,20,17,15,14,11,10,8,6,4), (1,17)(2,18)(3,4)(5,14)(6,13)(7,23)(8,24)(11,20)(12,19)(15,16)
$|\Aut(F/K)|$:  $4$

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:  $D_{6}$
16:  $C_8:C_2$
24:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Degree 8: $C_8:C_2$

Degree 12: $S_3 \times C_4$

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 5]
Character table:   
      2  4  3  4   3  3   3   3  3   3  3  3  3  3  4  4   3   3  3
      3  1  .  1   1  .   1   1  .   1  1  .  1  1  1  1   1   1  1

        1a 2a 2b 24a 8a 24b 12a 4a 12b 8b 8c 3a 6a 4b 4c 24c 24d 8d
     2P 1a 1a 1a 12b 4c 12b  6a 2b  6a 4c 4b 3a 3a 2b 2b 12a 12a 4b
     3P 1a 2a 2b  8b 8c  8b  4b 4a  4c 8d 8a 1a 2b 4c 4b  8d  8d 8b
     5P 1a 2a 2b 24a 8a 24b 12a 4a 12b 8b 8c 3a 6a 4b 4c 24c 24d 8d
     7P 1a 2a 2b 24d 8c 24c 12b 4a 12a 8d 8a 3a 6a 4c 4b 24b 24a 8b
    11P 1a 2a 2b 24d 8c 24c 12b 4a 12a 8d 8a 3a 6a 4c 4b 24b 24a 8b
    13P 1a 2a 2b 24b 8a 24a 12a 4a 12b 8b 8c 3a 6a 4b 4c 24d 24c 8d
    17P 1a 2a 2b 24b 8a 24a 12a 4a 12b 8b 8c 3a 6a 4b 4c 24d 24c 8d
    19P 1a 2a 2b 24c 8c 24d 12b 4a 12a 8d 8a 3a 6a 4c 4b 24a 24b 8b
    23P 1a 2a 2b 24c 8c 24d 12b 4a 12a 8d 8a 3a 6a 4c 4b 24a 24b 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   A -A   A  -1  1  -1 -A  A  1  1 -1 -1  -A  -A  A
X.6      1 -1  1  -A  A  -A  -1  1  -1  A -A  1  1 -1 -1   A   A -A
X.7      1  1  1   A  A   A  -1 -1  -1 -A -A  1  1 -1 -1  -A  -A  A
X.8      1  1  1  -A -A  -A  -1 -1  -1  A  A  1  1 -1 -1   A   A -A
X.9      2  .  2  -1  .  -1  -1  .  -1  2  . -1 -1  2  2  -1  -1  2
X.10     2  .  2   1  .   1  -1  .  -1 -2  . -1 -1  2  2   1   1 -2
X.11     2  . -2   .  .   .   C  .  -C  .  .  2 -2 -C  C   .   .  .
X.12     2  . -2   .  .   .  -C  .   C  .  .  2 -2  C -C   .   .  .
X.13     2  . -2   B  .  -B   A  .  -A  .  . -1  1  C -C  /B -/B  .
X.14     2  . -2 -/B  .  /B  -A  .   A  .  . -1  1 -C  C  -B   B  .
X.15     2  . -2  /B  . -/B  -A  .   A  .  . -1  1 -C  C   B  -B  .
X.16     2  . -2  -B  .   B   A  .  -A  .  . -1  1  C -C -/B  /B  .
X.17     2  .  2   A  .   A   1  .   1  C  . -1 -1 -2 -2  -A  -A -C
X.18     2  .  2  -A  .  -A   1  .   1 -C  . -1 -1 -2 -2   A   A  C

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(24)+E(24)^17
C = -2*E(4)
  = -2*Sqrt(-1) = -2i