Properties

Label 24T5
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:Q_8$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $5$
Group :  $C_3:Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,2,19)(3,17,4,18)(5,15,6,16)(7,13,8,14)(9,12,10,11)(21,23,22,24), (1,7,18,23,9,15,2,8,17,24,10,16)(3,13,20,5,11,21,4,14,19,6,12,22)
$|\Aut(F/K)|$:  $24$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $Q_8$
12:  $D_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

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

Degree 8: $Q_8$

Degree 12: $D_6$

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 4]
Character table:   
     2  3  3  2  2   2   2  2  2  2
     3  1  1  .  .   1   1  1  1  1

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

X.1     1  1  1  1   1   1  1  1  1
X.2     1  1 -1 -1   1   1  1  1  1
X.3     1  1 -1  1  -1  -1  1  1 -1
X.4     1  1  1 -1  -1  -1  1  1 -1
X.5     2 -2  .  .   .   .  2 -2  .
X.6     2  2  .  .  -1  -1 -1 -1  2
X.7     2  2  .  .   1   1 -1 -1 -2
X.8     2 -2  .  .   A  -A -1  1  .
X.9     2 -2  .  .  -A   A -1  1  .

A = -E(12)^7+E(12)^11
  = Sqrt(3) = r3