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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);
 
Nilpotency class:  $-1$ (not nilpotent)
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

|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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $4$ $3$ $( 3, 6,15)( 4, 5,16)( 7,14, 9)( 8,13,10)$
$ 3, 3, 3, 3, 1, 1, 1, 1 $ $4$ $3$ $( 3,15, 6)( 4,16, 5)( 7, 9,14)( 8,10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 6, 6, 2, 2 $ $4$ $6$ $( 1, 2)( 3, 5,15, 4, 6,16)( 7,13, 9, 8,14,10)(11,12)$
$ 6, 6, 2, 2 $ $4$ $6$ $( 1, 2)( 3,16, 6, 4,15, 5)( 7,10,14, 8, 9,13)(11,12)$
$ 4, 4, 4, 4 $ $6$ $4$ $( 1, 3, 2, 4)( 5,16, 6,15)( 7,14, 8,13)( 9,12,10,11)$
$ 12, 4 $ $4$ $12$ $( 1, 7,15,12, 6,13, 2, 8,16,11, 5,14)( 3, 9, 4,10)$
$ 12, 4 $ $4$ $12$ $( 1, 7, 4,11, 5,10, 2, 8, 3,12, 6, 9)(13,16,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 7)( 2, 8)( 3,13)( 4,14)( 5,11)( 6,12)( 9,15)(10,16)$
$ 12, 4 $ $4$ $12$ $( 1, 8,15,11, 6,14, 2, 7,16,12, 5,13)( 3,10, 4, 9)$
$ 12, 4 $ $4$ $12$ $( 1, 8, 4,12, 5, 9, 2, 7, 3,11, 6,10)(13,15,14,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5, 8, 6, 7)(13,15,14,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3,10, 4, 9)( 5, 7, 6, 8)(13,16,14,15)$

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);
 
Label:  48.33
magma: IdentifyGroup(G);
 
Character table:    
      2  4   2   2  4  2  2  3   2   2  3   2   2  4  4
      3  1   1   1  1  1  1  .   1   1  .   1   1  1  1

        1a  3a  3b 2a 6a 6b 4a 12a 12b 2b 12c 12d 4b 4c
     2P 1a  3b  3a 1a 3b 3a 2a  6a  6b 1a  6a  6b 2a 2a
     3P 1a  1a  1a 2a 2a 2a 4a  4c  4b 2b  4b  4c 4c 4b
     5P 1a  3b  3a 2a 6b 6a 4a 12d 12c 2b 12b 12a 4b 4c
     7P 1a  3a  3b 2a 6a 6b 4a 12c 12d 2b 12a 12b 4c 4b
    11P 1a  3b  3a 2a 6b 6a 4a 12b 12a 2b 12d 12c 4c 4b

X.1      1   1   1  1  1  1  1   1   1  1   1   1  1  1
X.2      1   1   1  1  1  1  1  -1  -1 -1  -1  -1 -1 -1
X.3      1   A  /A  1  A /A  1 -/A  -A -1 -/A  -A -1 -1
X.4      1  /A   A  1 /A  A  1  -A -/A -1  -A -/A -1 -1
X.5      1   A  /A  1  A /A  1  /A   A  1  /A   A  1  1
X.6      1  /A   A  1 /A  A  1   A  /A  1   A  /A  1  1
X.7      2  -1  -1 -2  1  1  .   B  -B  .  -B   B  D -D
X.8      2  -1  -1 -2  1  1  .  -B   B  .   B  -B -D  D
X.9      2 -/A  -A -2 /A  A  .   C  /C  .  -C -/C  D -D
X.10     2 -/A  -A -2 /A  A  .  -C -/C  .   C  /C -D  D
X.11     2  -A -/A -2  A /A  . -/C  -C  .  /C   C  D -D
X.12     2  -A -/A -2  A /A  .  /C   C  . -/C  -C -D  D
X.13     3   .   .  3  .  . -1   .   . -1   .   .  3  3
X.14     3   .   .  3  .  . -1   .   .  1   .   . -3 -3

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(12)^11
D = 2*E(4)
  = 2*Sqrt(-1) = 2i

magma: CharacterTable(G);