Properties

Label 24T8
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:C_8$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $8$
Group :  $C_3:C_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,13,20,2,7,14,19)(3,17,16,6,4,18,15,5)(9,23,21,12,10,24,22,11), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,22,14)(6,21,13)(7,24,15)(8,23,16)
$|\Aut(F/K)|$:  $24$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
8:  $C_8$
12:  $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $S_3$

Degree 8: $C_8$

Degree 12: $C_3 : C_4$

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

Group invariants

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

        1a 2a  8a  8b 12a 12b  8c  8d 3a 6a 4a 4b
     2P 1a 1a  4b  4b  6a  6a  4a  4a 3a 3a 2a 2a
     3P 1a 2a  8c  8d  4a  4b  8a  8b 1a 2a 4b 4a
     5P 1a 2a  8b  8a 12a 12b  8d  8c 3a 6a 4a 4b
     7P 1a 2a  8d  8c 12b 12a  8b  8a 3a 6a 4b 4a
    11P 1a 2a  8c  8d 12b 12a  8a  8b 3a 6a 4b 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   A  -A  -B   B -/A  /A  1 -1  B -B
X.4      1 -1 -/A  /A   B  -B   A  -A  1 -1 -B  B
X.5      1 -1  /A -/A   B  -B  -A   A  1 -1 -B  B
X.6      1 -1  -A   A  -B   B  /A -/A  1 -1  B -B
X.7      1  1   B   B  -1  -1  -B  -B  1  1 -1 -1
X.8      1  1  -B  -B  -1  -1   B   B  1  1 -1 -1
X.9      2  2   .   .  -1  -1   .   . -1 -1  2  2
X.10     2  2   .   .   1   1   .   . -1 -1 -2 -2
X.11     2 -2   .   .   B  -B   .   . -1  1  C -C
X.12     2 -2   .   .  -B   B   .   . -1  1 -C  C

A = -E(8)
B = -E(4)
  = -Sqrt(-1) = -i
C = -2*E(4)
  = -2*Sqrt(-1) = -2i