Properties

Label 16T1
Order \(16\)
n \(16\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_{16}$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1$
Group :  $C_{16}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,7,14,4,9,15,5,11,2,8,13,3,10,16,6,12)
$|\Aut(F/K)|$:  $16$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

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

Group invariants

Order:  $16=2^{4}$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [16, 1]
Character table:   
      2  4  4   4   4   4   4   4   4  4  4   4   4   4   4   4   4

        1a 2a 16a 16b  8a  8b 16c 16d 4a 4b 16e 16f  8c  8d 16g 16h

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   A   A  -1  -1  -A  -A  1  1   A   A  -1  -1  -A  -A
X.4      1  1  -A  -A  -1  -1   A   A  1  1  -A  -A  -1  -1   A   A
X.5      1  1   B   B   A   A -/B -/B -1 -1  -B  -B  -A  -A  /B  /B
X.6      1  1  -B  -B   A   A  /B  /B -1 -1   B   B  -A  -A -/B -/B
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   C  -C   B  -B -/D  /D -A  A   D  -D  /B -/B  /C -/C
X.10     1 -1  -C   C   B  -B  /D -/D -A  A  -D   D  /B -/B -/C  /C
X.11     1 -1   D  -D  -B   B  /C -/C -A  A  -C   C -/B  /B  /D -/D
X.12     1 -1  -D   D  -B   B -/C  /C -A  A   C  -C -/B  /B -/D  /D
X.13     1 -1 -/D  /D -/B  /B  -C   C  A -A  /C -/C  -B   B  -D   D
X.14     1 -1  /D -/D -/B  /B   C  -C  A -A -/C  /C  -B   B   D  -D
X.15     1 -1 -/C  /C  /B -/B   D  -D  A -A -/D  /D   B  -B  -C   C
X.16     1 -1  /C -/C  /B -/B  -D   D  A -A  /D -/D   B  -B   C  -C

A = -E(4)
  = -Sqrt(-1) = -i
B = E(8)^3
C = -E(16)^3
D = E(16)^7