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Properties

Label 16T60
Degree $16$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\SL(2,3):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 60);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $60$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\SL(2,3):C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,5,11,16,8,2,13,6,12,15,7)(3,10,4,9), (1,13)(2,14)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$

Degree 8: $A_4\times C_2$

Low degree siblings

24T21

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{8}$ $6$ $2$ $8$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5,10)( 6, 9)(11,15)(12,16)$
3A1 $3^{4},1^{4}$ $4$ $3$ $8$ $( 3, 6,15)( 4, 5,16)( 7,14, 9)( 8,13,10)$
3A-1 $3^{4},1^{4}$ $4$ $3$ $8$ $( 3,15, 6)( 4,16, 5)( 7, 9,14)( 8,10,13)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 8, 6, 7)(13,15,14,16)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$
4B $4^{4}$ $6$ $4$ $12$ $( 1, 5, 2, 6)( 3,15, 4,16)( 7,11, 8,12)( 9,14,10,13)$
6A1 $6^{2},2^{2}$ $4$ $6$ $12$ $( 1, 4, 5, 2, 3, 6)( 7,11,10, 8,12, 9)(13,14)(15,16)$
6A-1 $6^{2},2^{2}$ $4$ $6$ $12$ $( 1, 6, 3, 2, 5, 4)( 7, 9,12, 8,10,11)(13,14)(15,16)$
12A1 $12,4$ $4$ $12$ $14$ $( 1, 9,16,11, 4,13, 2,10,15,12, 3,14)( 5, 7, 6, 8)$
12A-1 $12,4$ $4$ $12$ $14$ $( 1,10,16,12, 4,14, 2, 9,15,11, 3,13)( 5, 8, 6, 7)$
12A5 $12,4$ $4$ $12$ $14$ $( 1,14, 3,12,15,10, 2,13, 4,11,16, 9)( 5, 8, 6, 7)$
12A-5 $12,4$ $4$ $12$ $14$ $( 1,13, 3,11,15, 9, 2,14, 4,12,16,10)( 5, 7, 6, 8)$

Malle's constant $a(G)$:     $1/8$

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:  48.33
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A1 3A-1 4A1 4A-1 4B 6A1 6A-1 12A1 12A-1 12A5 12A-5
Size 1 1 6 4 4 1 1 6 4 4 4 4 4 4
2 P 1A 1A 1A 3A-1 3A1 2A 2A 2A 3A1 3A-1 6A1 6A1 6A-1 6A-1
3 P 1A 2A 2B 1A 1A 4A-1 4A1 4B 2A 2A 4A1 4A-1 4A-1 4A1
Type
48.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.33.1b R 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1
48.33.1c1 C 1 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
48.33.1c2 C 1 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
48.33.1d1 C 1 1 −1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.33.1d2 C 1 1 −1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.33.2a1 C 2 −2 0 −1 −1 −2i 2i 0 1 1 −i i −i i
48.33.2a2 C 2 −2 0 −1 −1 2i −2i 0 1 1 i −i i −i
48.33.2b1 C 2 −2 0 −ζ124 ζ122 −2ζ123 2ζ123 0 −ζ122 ζ124 ζ12 −ζ125 ζ125 −ζ12
48.33.2b2 C 2 −2 0 ζ122 −ζ124 2ζ123 −2ζ123 0 ζ124 −ζ122 −ζ125 ζ12 −ζ12 ζ125
48.33.2b3 C 2 −2 0 −ζ124 ζ122 2ζ123 −2ζ123 0 −ζ122 ζ124 −ζ12 ζ125 −ζ125 ζ12
48.33.2b4 C 2 −2 0 ζ122 −ζ124 −2ζ123 2ζ123 0 ζ124 −ζ122 ζ125 −ζ12 ζ12 −ζ125
48.33.3a R 3 3 −1 0 0 3 3 −1 0 0 0 0 0 0
48.33.3b R 3 3 1 0 0 −3 −3 −1 0 0 0 0 0 0

magma: CharacterTable(G);