Properties

Label 24T36
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $Q_8:S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $36$
Group :  $Q_8:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,18,20,9,11,2,4,17,19,10,12)(5,8,21,24,14,16,6,7,22,23,13,15), (3,23)(4,24)(5,21)(6,22)(7,20)(8,19)(9,17)(10,18)(11,16)(12,15)(13,14)
$|\Aut(F/K)|$:  $2$

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:  $QD_{16}$
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: $QD_{16}$

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

Low degree siblings

There are no siblings 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 $ $6$ $8$ $( 1, 3,13,16, 2, 4,14,15)( 5,24,17,11, 6,23,18,12)( 7, 9,19,21, 8,10,20,22)$
$ 12, 12 $ $4$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
$ 8, 8, 8 $ $6$ $8$ $( 1, 4,13,15, 2, 3,14,16)( 5,23,17,12, 6,24,18,11)( 7,10,19,22, 8, 9,20,21)$
$ 12, 12 $ $4$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
$ 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3,21, 4,22)( 5,11, 6,12)( 9,15,10,16)(13,20,14,19)(17,24,18,23)$
$ 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 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 12, 12 $ $4$ $12$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,16,13,24,22, 8, 6,15,14,23,21, 7)$
$ 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, 17]
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 12a 8b 12b 4a 3a 6a 12c 4b
     2P 1a 1a 1a 4b  6a 4b  6a 2b 3a 3a  6a 2b
     3P 1a 2a 2b 8a  4a 8b  4b 4a 1a 2b  4a 4b
     5P 1a 2a 2b 8b 12c 8a 12b 4a 3a 6a 12a 4b
     7P 1a 2a 2b 8b 12a 8a 12b 4a 3a 6a 12c 4b
    11P 1a 2a 2b 8a 12c 8b 12b 4a 3a 6a 12a 4b

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) = -i2
B = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3