Properties

Label 24T26
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times Q_8$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $26$
Group :  $S_3\times Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23), (1,14,2,13)(3,24,4,23)(5,10,6,9)(7,20,8,19)(11,15,12,16)(17,22,18,21), (1,24,10,8,17,15,2,23,9,7,18,16)(3,13,12,22,19,6,4,14,11,21,20,5)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$, $Q_8$ x 2
12:  $D_{6}$ x 3
16:  $D_8$
24:  $S_3 \times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 8: $Q_8$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T26

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 40]
Character table:   
      2  4  4  4  4  3   2   2  3  3  3  3  3  3   2  3
      3  1  .  1  .  .   1   1  .  .  1  1  1  1   1  1

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

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  1 -1  1  -1   1 -1  1 -1  1  1  1  -1 -1
X.6      1  1  1  1 -1  -1  -1 -1  1  1  1  1 -1   1 -1
X.7      1  1  1  1 -1  -1   1  1 -1 -1  1  1  1  -1 -1
X.8      1  1  1  1  1   1  -1 -1 -1 -1  1  1 -1  -1  1
X.9      2 -2 -2  2  .   .   .  .  .  .  2 -2  .   .  .
X.10     2  2 -2 -2  .   .   .  .  .  .  2 -2  .   .  .
X.11     2  .  2  .  .  -1  -1  .  .  2 -1 -1  2  -1  2
X.12     2  .  2  .  .  -1   1  .  . -2 -1 -1 -2   1  2
X.13     2  .  2  .  .   1  -1  .  . -2 -1 -1  2   1 -2
X.14     2  .  2  .  .   1   1  .  .  2 -1 -1 -2  -1 -2
X.15     4  . -4  .  .   .   .  .  .  . -2  2  .   .  .