Properties

Label 24T25
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:D_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $25$
Group :  $C_2\times C_3:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,23,2,24)(3,22,4,21)(5,20,6,19)(7,17,8,18)(9,16,10,15)(11,14,12,13), (1,20)(2,19)(3,10)(4,9)(5,23)(6,24)(7,13)(8,14)(11,18)(12,17)(15,21)(16,22)
$|\Aut(F/K)|$:  $4$

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:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 3
16:  $D_4\times C_2$
24:  $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $D_{6}$ x 3

Degree 8: $D_4\times C_2$

Degree 12: $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2

Low degree siblings

24T25, 24T45 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 $ $6$ $2$ $( 3,12)( 4,11)( 5,22)( 6,21)( 7, 8)( 9,17)(10,18)(15,23)(16,24)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 3, 2, 4)( 5,24, 6,23)( 7,21, 8,22)( 9,19,10,20)(11,18,12,17)(13,16,14,15)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 3,17,19, 9,11)( 2, 4,18,20,10,12)( 5, 7,22,24,14,15)( 6, 8,21,23,13,16)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 4,17,20, 9,12)( 2, 3,18,19,10,11)( 5, 8,22,23,14,16)( 6, 7,21,24,13,15)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,22)(10,21)(13,18)(14,17)(19,23) (20,24)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
$ 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 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,22)( 4,21)( 5,11)( 6,12)( 9,15)(10,16)(13,20)(14,19)(17,24) (18,23)$
$ 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 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,20)( 8,19)( 9,21)(10,22)(11,23) (12,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,22)(10,21)(11,24) (12,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,15,17, 7, 9,24)( 2,16,18, 8,10,23)( 3, 5,19,22,11,14)( 4, 6,20,21,12,13)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 6,19,21,11,13)( 4, 5,20,22,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,19)( 2,20)( 3, 9)( 4,10)( 5,24)( 6,23)( 7,14)( 8,13)(11,17)(12,18)(15,22) (16,21)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 43]
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 4a 6a 6b 6c 2c 6d 4b 2d 3a 6e 2e 2f 6f 6g 2g
     2P 1a 1a 1a 2b 3a 3a 3a 1a 3a 2b 1a 3a 3a 1a 1a 3a 3a 1a
     3P 1a 2a 2b 4a 2g 2g 2f 2c 2e 4b 2d 1a 2b 2e 2f 2d 2d 2g
     5P 1a 2a 2b 4a 6b 6a 6c 2c 6d 4b 2d 3a 6e 2e 2f 6g 6f 2g

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  1 -1 -1 -1
X.6      1  1  1 -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 -1 -1 -1
X.8      1  1  1  1  1  1 -1 -1 -1 -1 -1  1  1 -1 -1 -1 -1  1
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  .  1  1 -1  . -1  . -2 -1 -1  2  2  1  1 -2
X.12     2  .  2  .  1  1  1  .  1  .  2 -1 -1 -2 -2 -1 -1 -2
X.13     2  . -2  .  .  . -2  .  2  .  .  2 -2  2 -2  .  .  .
X.14     2  . -2  .  .  .  2  . -2  .  .  2 -2 -2  2  .  .  .
X.15     2  . -2  .  A -A -1  .  1  .  . -1  1 -2  2  A -A  .
X.16     2  . -2  . -A  A -1  .  1  .  . -1  1 -2  2 -A  A  .
X.17     2  . -2  .  A -A  1  . -1  .  . -1  1  2 -2 -A  A  .
X.18     2  . -2  . -A  A  1  . -1  .  . -1  1  2 -2  A -A  .

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3