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Properties

Label 24T23
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4:S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $23$
Group :  $D_4:S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,2,22)(3,8,4,7)(5,17,6,18)(9,13,10,14)(11,23,12,24)(15,19,16,20), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22), (3,11)(4,12)(5,21)(6,22)(7,8)(9,17)(10,18)(13,14)(15,23)(16,24)
$|\Aut(F/K)|$:  $4$

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:  $C_2^3$
12:  $D_{6}$ x 3
16:  $Q_8:C_2$
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: $Q_8:C_2$

Degree 12: $S_3 \times C_2^2$

Low degree siblings

24T18 x 2

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, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 3,11)( 4,12)( 5,21)( 6,22)( 7, 8)( 9,17)(10,18)(13,14)(15,23)(16,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 3, 2, 4)( 5,24, 6,23)( 7,21, 8,22)( 9,19,10,20)(11,18,12,17)(13,16,14,15)$
$ 12, 12 $ $4$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 4, 2, 3)( 5,23, 6,24)( 7,22, 8,21)( 9,20,10,19)(11,17,12,18)(13,15,14,16)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
$ 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 $ $6$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,21,12,22)(13,20,14,19)(15,18,16,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,21)( 4,22)( 5,12)( 6,11)( 9,15)(10,16)(13,19)(14,20)(17,24) (18,23)$
$ 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 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,21)(10,22)(11,23) (12,24)$
$ 6, 6, 6, 6 $ $4$ $6$ $( 1,15,17, 7, 9,24)( 2,16,18, 8,10,23)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
$ 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, 39]
Character table:    
      2  4  3  4  4   2  4  2  3  3  3  3  3  3  2  3
      3  1  .  1  .   1  .  1  .  .  1  1  1  1  1  1

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

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  .  -1  . -1  .  .  2 -1 -1  2 -1  2
X.10     2  .  2  .  -1  .  1  .  . -2 -1 -1 -2  1  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  A   . -A  .  .  .  .  2 -2  .  .  .
X.14     2  . -2 -A   .  A  .  .  .  .  2 -2  .  .  .
X.15     4  . -4  .   .  .  .  .  .  . -2  2  .  .  .

A = -2*E(4)
  = -2*Sqrt(-1) = -2i