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Properties

Label 24T43
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:D_8$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $43$
Group :  $C_3:D_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,7)(3,6)(4,5)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,18)(16,17), (1,6,9,13,17,21)(2,5,10,14,18,22)(3,20,11,4,19,12)(7,24,15)(8,23,16)
$|\Aut(F/K)|$:  $6$

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:  $D_{8}$
24:  $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Degree 8: $D_{8}$

Degree 12: $(C_6\times C_2):C_2$

Low degree siblings

24T37

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 15]
Character table:    
      2  4  2  4  2  3  3   2  2  3  3  3  2
      3  1  1  1  .  .  .   1  1  1  1  1  1

        1a 2a 2b 2c 8a 8b 12a 6a 3a 6b 4a 6c
     2P 1a 1a 1a 1a 4a 4a  6b 3a 3a 3a 2b 3a
     3P 1a 2a 2b 2c 8b 8a  4a 2a 1a 2b 4a 2a
     5P 1a 2a 2b 2c 8b 8a 12a 6c 3a 6b 4a 6a
     7P 1a 2a 2b 2c 8a 8b 12a 6a 3a 6b 4a 6c
    11P 1a 2a 2b 2c 8b 8a 12a 6c 3a 6b 4a 6a

X.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
X.3      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
X.5      2  .  2  .  .  .  -2  .  2  2 -2  .
X.6      2 -2  2  .  .  .  -1  1 -1 -1  2  1
X.7      2  2  2  .  .  .  -1 -1 -1 -1  2 -1
X.8      2  . -2  .  A -A   .  .  2 -2  .  .
X.9      2  . -2  . -A  A   .  .  2 -2  .  .
X.10     2  .  2  .  .  .   1  B -1 -1 -2 -B
X.11     2  .  2  .  .  .   1 -B -1 -1 -2  B
X.12     4  . -4  .  .  .   .  . -2  2  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3