Properties

Label 24T35
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_8:S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $35$
Group :  $C_8:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13)(2,14)(3,12)(4,11)(5,10)(6,9)(15,24)(16,23)(17,21)(18,22)(19,20), (1,24,2,23)(3,21,4,22)(5,19,6,20)(7,18,8,17)(9,15,10,16)(11,13,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
16:  $QD_{16}$
24:  $D_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: $QD_{16}$

Degree 12: $D_{12}$

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 6]
Character table:   
      2  4  2  4  2   3   3   3   3  3  3  3  3   3   3  3
      3  1  .  1  .   1   1   1   1  1  1  1  1   1   1  1

        1a 2a 2b 4a 24a 24b 12a 12b 8a 8b 3a 6a 24c 24d 4b
     2P 1a 1a 1a 2b 12b 12b  6a  6a 4b 4b 3a 3a 12a 12a 2b
     3P 1a 2a 2b 4a  8b  8a  4b  4b 8a 8b 1a 2b  8b  8a 4b
     5P 1a 2a 2b 4a 24d 24c 12b 12a 8b 8a 3a 6a 24b 24a 4b
     7P 1a 2a 2b 4a 24d 24c 12b 12a 8b 8a 3a 6a 24b 24a 4b
    11P 1a 2a 2b 4a 24a 24b 12a 12b 8a 8b 3a 6a 24c 24d 4b
    13P 1a 2a 2b 4a 24b 24a 12a 12b 8b 8a 3a 6a 24d 24c 4b
    17P 1a 2a 2b 4a 24c 24d 12b 12a 8a 8b 3a 6a 24a 24b 4b
    19P 1a 2a 2b 4a 24c 24d 12b 12a 8a 8b 3a 6a 24a 24b 4b
    23P 1a 2a 2b 4a 24b 24a 12a 12b 8b 8a 3a 6a 24d 24c 4b

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      2  .  2  .   .   .  -2  -2  .  .  2  2   .   . -2
X.6      2  .  2  .   1   1  -1  -1 -2 -2 -1 -1   1   1  2
X.7      2  .  2  .  -1  -1  -1  -1  2  2 -1 -1  -1  -1  2
X.8      2  . -2  .   A  -A   .   . -A  A  2 -2   A  -A  .
X.9      2  . -2  .  -A   A   .   .  A -A  2 -2  -A   A  .
X.10     2  . -2  .   B  -B  -D   D -A  A -1  1   C  -C  .
X.11     2  . -2  .   C  -C   D  -D -A  A -1  1   B  -B  .
X.12     2  . -2  .  -C   C   D  -D  A -A -1  1  -B   B  .
X.13     2  . -2  .  -B   B  -D   D  A -A -1  1  -C   C  .
X.14     2  .  2  .   D   D   1   1  .  . -1 -1  -D  -D -2
X.15     2  .  2  .  -D  -D   1   1  .  . -1 -1   D   D -2

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
B = -E(24)-E(24)^11
C = -E(24)^17-E(24)^19
D = -E(12)^7+E(12)^11
  = Sqrt(3) = r3