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Properties

Label 24T9
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 9);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $24$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  24.13
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);