Properties

Label 24T9
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times A_4$

Learn more about

Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $9$
Group :  $C_2\times A_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,9)(4,10)(5,11)(6,12)(7,8)(13,14)(15,21)(16,22)(17,23)(18,24)(19,20), (1,3,11,14,22,17)(2,4,12,13,21,18)(5,8,10,23,19,15)(6,7,9,24,20,16)
$|\Aut(F/K)|$:  $24$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $A_4$

Degree 6: $C_6$, $A_4$, $A_4\times C_2$

Degree 8: $A_4\times C_2$

Degree 12: $A_4$, $A_4\times C_2$, $A_4 \times C_2$

Low degree siblings

6T6, 8T13, 12T6, 12T7

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 13]
Character table:   
     2  3  3   1   1   1   1  3  3
     3  1  .   1   1   1   1  .  1

       1a 2a  6a  3a  6b  3b 2b 2c
    2P 1a 1a  3b  3b  3a  3a 1a 1a
    3P 1a 2a  2c  1a  2c  1a 2b 2c
    5P 1a 2a  6b  3b  6a  3a 2b 2c

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

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3