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Properties

Label 24T50
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 50);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\times A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,18,13,10,6)(2,21,17,14,9,5)(3,23,20,15,11,8)(4,24,19,16,12,7), (1,4,10,23,5,8)(2,3,9,24,6,7)(11,17,20,13,16,22)(12,18,19,14,15,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3, $A_4$, $A_4\times C_2$ x 3

Degree 8: None

Degree 12: $C_6\times C_2$, $A_4 \times C_2$ x 3, $C_2^2 \times A_4$ x 3

Low degree siblings

12T25 x 3, 12T26 x 2, 16T58, 24T49 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1
Size 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1
3 P 1A 2C 2A 2B 2G 2D 2F 2E 1A 1A 2B 2A 2B 2A 2C 2C
Type
48.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.49.1b R 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
48.49.1c R 1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
48.49.1d R 1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1
48.49.1e1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
48.49.1e2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
48.49.1f1 C 1 −1 −1 1 −1 −1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3
48.49.1f2 C 1 −1 −1 1 −1 −1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1
48.49.1g1 C 1 −1 1 −1 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3
48.49.1g2 C 1 −1 1 −1 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1
48.49.1h1 C 1 1 −1 −1 −1 1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
48.49.1h2 C 1 1 −1 −1 −1 1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1
48.49.3a R 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0
48.49.3b R 3 −3 −3 3 1 1 −1 −1 0 0 0 0 0 0 0 0
48.49.3c R 3 −3 3 −3 −1 1 1 −1 0 0 0 0 0 0 0 0
48.49.3d R 3 3 −3 −3 1 −1 1 −1 0 0 0 0 0 0 0 0

magma: CharacterTable(G);