Properties

Label 15T8
Order \(60\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_5\times C_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $8$
Group :  $F_5\times C_3$
CHM label :  $F(5)[x]3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,4,13)(2,14,8,11)(3,6,12,9), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $F_5$

Low degree siblings

30T7

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

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [60, 6]
Character table:   
      2  2  2  2  2   .   2   2   2   2   .   2   2  .  2  2
      3  1  1  1  1   1   1   1   1   1   1   1   1  1  1  1
      5  1  .  .  .   1   .   .   .   .   1   .   .  1  1  1

        1a 2a 4a 4b 15a  6a 12a 12b 12c 15b  6b 12d 5a 3a 3b
     2P 1a 1a 2a 2a 15b  3a  6b  6b  6a 15a  3b  6a 5a 3b 3a
     3P 1a 2a 4b 4a  5a  2a  4b  4a  4a  5a  2a  4b 5a 1a 1a
     5P 1a 2a 4a 4b  3a  6b 12d 12c 12b  3b  6a 12a 1a 3b 3a
     7P 1a 2a 4b 4a 15a  6a 12b 12a 12d 15b  6b 12c 5a 3a 3b
    11P 1a 2a 4b 4a 15b  6b 12c 12d 12a 15a  6a 12b 5a 3b 3a
    13P 1a 2a 4a 4b 15a  6a 12a 12b 12c 15b  6b 12d 5a 3a 3b

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

A = -E(4)
  = -Sqrt(-1) = -i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = -E(12)^11
D = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3