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Properties

Label 15T1
Order \(15\)
n \(15\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{15}$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
5:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

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

Group invariants

Order:  $15=3 \cdot 5$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [15, 1]
Character table:    
      3  1   1   1  1   1  1  1   1   1  1  1   1  1   1   1
      5  1   1   1  1   1  1  1   1   1  1  1   1  1   1   1

        1a 15a 15b 5a 15c 3a 5b 15d 15e 5c 3b 15f 5d 15g 15h

X.1      1   1   1  1   1  1  1   1   1  1  1   1  1   1   1
X.2      1   A  /A  1   A /A  1   A  /A  1  A  /A  1   A  /A
X.3      1  /A   A  1  /A  A  1  /A   A  1 /A   A  1  /A   A
X.4      1   B   E /E  /B  1  B   E  /E /B  1   B  E  /E  /B
X.5      1   C   G /E  /D /A  B   F  /F /B  A   D  E  /G  /C
X.6      1   D   F /E  /C  A  B   G  /G /B /A   C  E  /F  /D
X.7      1   E  /B  B  /E  1  E  /B   B /E  1   E /B   B  /E
X.8      1   F  /C  B  /G /A  E  /D   D /E  A   G /B   C  /F
X.9      1   G  /D  B  /F  A  E  /C   C /E /A   F /B   D  /G
X.10     1  /E   B /B   E  1 /E   B  /B  E  1  /E  B  /B   E
X.11     1  /G   D /B   F /A /E   C  /C  E  A  /F  B  /D   G
X.12     1  /F   C /B   G  A /E   D  /D  E /A  /G  B  /C   F
X.13     1  /B  /E  E   B  1 /B  /E   E  B  1  /B /E   E   B
X.14     1  /D  /F  E   C /A /B  /G   G  B  A  /C /E   F   D
X.15     1  /C  /G  E   D  A /B  /F   F  B /A  /D /E   G   C

A = E(3)
  = (-1+Sqrt(-3))/2 = b3
B = E(5)^2
C = E(15)^11
D = E(15)
E = E(5)^4
F = E(15)^2
G = E(15)^7