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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 invariants

Abstract group:  $C_2^2\times A_4$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{12}$ $1$ $2$ $12$ $( 1,23)( 2,24)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,18)(16,17)(19,21)(20,22)$
2B $2^{12}$ $1$ $2$ $12$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22)(11,23)(12,24)$
2C $2^{12}$ $1$ $2$ $12$ $( 1,11)( 2,12)( 3,18)( 4,17)( 5,16)( 6,15)( 7,21)( 8,22)( 9,19)(10,20)(13,23)(14,24)$
2D $2^{12}$ $3$ $2$ $12$ $( 1,24)( 2,23)( 3, 5)( 4, 6)( 7,21)( 8,22)( 9,19)(10,20)(11,14)(12,13)(15,17)(16,18)$
2E $2^{8},1^{8}$ $3$ $2$ $8$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,18)( 6,17)(11,24)(12,23)$
2F $2^{12}$ $3$ $2$ $12$ $( 1,12)( 2,11)( 3,17)( 4,18)( 5,15)( 6,16)( 7, 9)( 8,10)(13,24)(14,23)(19,21)(20,22)$
2G $2^{12}$ $3$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14)(15,16)(17,18)(23,24)$
3A1 $3^{8}$ $4$ $3$ $16$ $( 1,10,18)( 2, 9,17)( 3,11,20)( 4,12,19)( 5,14,21)( 6,13,22)( 7,16,24)( 8,15,23)$
3A-1 $3^{8}$ $4$ $3$ $16$ $( 1,18,10)( 2,17, 9)( 3,20,11)( 4,19,12)( 5,21,14)( 6,22,13)( 7,24,16)( 8,23,15)$
6A1 $6^{4}$ $4$ $6$ $20$ $( 1,15,10,23,18, 8)( 2,16, 9,24,17, 7)( 3,22,11, 6,20,13)( 4,21,12, 5,19,14)$
6A-1 $6^{4}$ $4$ $6$ $20$ $( 1, 8,18,23,10,15)( 2, 7,17,24, 9,16)( 3,13,20, 6,11,22)( 4,14,19, 5,12,21)$
6B1 $6^{4}$ $4$ $6$ $20$ $( 1,22,18,13,10, 6)( 2,21,17,14, 9, 5)( 3,23,20,15,11, 8)( 4,24,19,16,12, 7)$
6B-1 $6^{4}$ $4$ $6$ $20$ $( 1, 6,10,13,18,22)( 2, 5, 9,14,17,21)( 3, 8,11,15,20,23)( 4, 7,12,16,19,24)$
6C1 $6^{4}$ $4$ $6$ $20$ $( 1, 3,10,11,18,20)( 2, 4, 9,12,17,19)( 5, 7,14,16,21,24)( 6, 8,13,15,22,23)$
6C-1 $6^{4}$ $4$ $6$ $20$ $( 1,20,18,11,10, 3)( 2,19,17,12, 9, 4)( 5,24,21,16,14, 7)( 6,23,22,15,13, 8)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(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 2A 2B 2C 2D 2E 2F 2G 1A 1A 2A 2A 2B 2B 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);
 

Regular extensions

Data not computed