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Properties

Label 24T45
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 $t$ :  $45$
Group :  $C_2\times C_3:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,23)(2,24)(3,21)(4,22)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,13)(12,14), (1,9,17)(2,10,18)(3,24,19,15,11,7)(4,23,20,16,12,8)(5,14,22)(6,13,21)
$|\Aut(F/K)|$:  $12$

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: $S_3$, $D_{6}$ x 2

Degree 8: $D_4\times C_2$

Degree 12: $D_6$, $(C_6\times C_2):C_2$ x 2

Low degree siblings

24T25 x 2, 24T45

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

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  3  4  4  3  3
      3  1  1  1  1  .  .  .  .  1  1  1  1  1  1  1  1  1  1

        1a 2a 2b 2c 2d 4a 2e 4b 6a 6b 6c 6d 3a 6e 2f 2g 6f 6g
     2P 1a 1a 1a 1a 1a 2g 1a 2g 3a 3a 3a 3a 3a 3a 1a 1a 3a 3a
     3P 1a 2a 2b 2c 2d 4a 2e 4b 2g 2a 2f 2c 1a 2b 2f 2g 2a 2c
     5P 1a 2a 2b 2c 2d 4a 2e 4b 6a 6f 6c 6g 3a 6e 2f 2g 6b 6d

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 -2  2  .  .  .  . -1  1  1 -1 -1  1 -2  2  1 -1
X.10     2 -2  2 -2  .  .  .  . -1  1 -1  1 -1 -1  2  2  1  1
X.11     2  2 -2 -2  .  .  .  . -1 -1  1  1 -1  1 -2  2 -1  1
X.12     2  2  2  2  .  .  .  . -1 -1 -1 -1 -1 -1  2  2 -1 -1
X.13     2  .  2  .  .  .  .  . -2  . -2  .  2  2 -2 -2  .  .
X.14     2  . -2  .  .  .  .  . -2  .  2  .  2 -2  2 -2  .  .
X.15     2  . -2  .  .  .  .  .  1  A -1 -A -1  1  2 -2 -A  A
X.16     2  . -2  .  .  .  .  .  1 -A -1  A -1  1  2 -2  A -A
X.17     2  .  2  .  .  .  .  .  1  A  1  A -1 -1 -2 -2 -A -A
X.18     2  .  2  .  .  .  .  .  1 -A  1 -A -1 -1 -2 -2  A  A

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