Properties

Label 24T37
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 $t$ :  $37$
Group :  $C_3:D_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,13,4,2,15,14,3)(5,11,17,23,6,12,18,24)(7,22,19,9,8,21,20,10), (1,15,17,7,9,24)(2,16,18,8,10,23)(3,5,19,22,11,14)(4,6,20,21,12,13)
$|\Aut(F/K)|$:  $2$

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: $D_{6}$

Degree 8: $D_{8}$

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

Low degree siblings

24T43

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

Group invariants

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

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

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

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