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Properties

Label 16T5
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_8\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 5);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_8\times C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $16$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,5,8,9,11,13,16)(2,3,6,7,10,12,14,15), (1,15,13,12,9,7,5,3)(2,16,14,11,10,8,6,4)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_8$ x 2, $C_4\times C_2$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 8, 8 $ $1$ $8$ $( 1, 3, 5, 7, 9,12,13,15)( 2, 4, 6, 8,10,11,14,16)$
$ 8, 8 $ $1$ $8$ $( 1, 4, 5, 8, 9,11,13,16)( 2, 3, 6, 7,10,12,14,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,12,15)( 4, 8,11,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,12,16)( 4, 7,11,15)$
$ 8, 8 $ $1$ $8$ $( 1, 7,13, 3, 9,15, 5,12)( 2, 8,14, 4,10,16, 6,11)$
$ 8, 8 $ $1$ $8$ $( 1, 8,13, 4, 9,16, 5,11)( 2, 7,14, 3,10,15, 6,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 8, 8 $ $1$ $8$ $( 1,11, 5,16, 9, 4,13, 8)( 2,12, 6,15,10, 3,14, 7)$
$ 8, 8 $ $1$ $8$ $( 1,12, 5,15, 9, 3,13, 7)( 2,11, 6,16,10, 4,14, 8)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,13, 9, 5)( 2,14,10, 6)( 3,15,12, 7)( 4,16,11, 8)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1,14, 9, 6)( 2,13,10, 5)( 3,16,12, 8)( 4,15,11, 7)$
$ 8, 8 $ $1$ $8$ $( 1,15,13,12, 9, 7, 5, 3)( 2,16,14,11,10, 8, 6, 4)$
$ 8, 8 $ $1$ $8$ $( 1,16,13,11, 9, 8, 5, 4)( 2,15,14,12,10, 7, 6, 3)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $16=2^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  16.5
magma: IdentifyGroup(G);
 
Character table:    
      2  4  4   4   4  4  4   4   4  4  4   4   4  4  4   4   4

        1a 2a  8a  8b 4a 4b  8c  8d 2b 2c  8e  8f 4c 4d  8g  8h
     2P 1a 1a  4a  4a 2b 2b  4c  4c 1a 1a  4a  4a 2b 2b  4c  4c
     3P 1a 2a  8c  8d 4c 4d  8a  8b 2b 2c  8h  8g 4a 4b  8f  8e
     5P 1a 2a  8f  8e 4a 4b  8g  8h 2b 2c  8b  8a 4c 4d  8c  8d
     7P 1a 2a  8g  8h 4c 4d  8f  8e 2b 2c  8d  8c 4a 4b  8a  8b

X.1      1  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  -1   1
X.3      1 -1   1  -1  1 -1   1  -1  1 -1  -1   1  1 -1   1  -1
X.4      1  1  -1  -1  1  1  -1  -1  1  1  -1  -1  1  1  -1  -1
X.5      1 -1   A  -A -1  1  -A   A  1 -1  -A   A -1  1  -A   A
X.6      1 -1  -A   A -1  1   A  -A  1 -1   A  -A -1  1   A  -A
X.7      1 -1   B  -B -A  A -/B  /B -1  1   B  -B  A -A  /B -/B
X.8      1 -1 -/B  /B  A -A   B  -B -1  1 -/B  /B -A  A  -B   B
X.9      1 -1  /B -/B  A -A  -B   B -1  1  /B -/B -A  A   B  -B
X.10     1 -1  -B   B -A  A  /B -/B -1  1  -B   B  A -A -/B  /B
X.11     1  1   A   A -1 -1  -A  -A  1  1   A   A -1 -1  -A  -A
X.12     1  1  -A  -A -1 -1   A   A  1  1  -A  -A -1 -1   A   A
X.13     1  1   B   B -A -A -/B -/B -1 -1  -B  -B  A  A  /B  /B
X.14     1  1 -/B -/B  A  A   B   B -1 -1  /B  /B -A -A  -B  -B
X.15     1  1  /B  /B  A  A  -B  -B -1 -1 -/B -/B -A -A   B   B
X.16     1  1  -B  -B -A -A  /B  /B -1 -1   B   B  A  A -/B -/B

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)

magma: CharacterTable(G);