Properties

Label 15T27
Order \(600\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $27$
CHM label :  $[5^{2}:4]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,7,4)(2,5,8,11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (1,7,4,13)(2,14,8,11)(3,6,12,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T27, 25T43, 30T150 x 2, 30T153 x 2, 30T155 x 2

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 $ $1$ $1$ $()$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 1,13,10, 7, 4)( 2, 5, 8,11,14)$
$ 5, 5, 5 $ $12$ $5$ $( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
$ 10, 5 $ $60$ $10$ $( 1,13,10, 7, 4)( 2,12, 5,15, 8, 3,11, 6,14, 9)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$
$ 6, 6, 3 $ $50$ $6$ $( 1,11, 3)( 2,12, 7, 5, 9,10)( 4, 8, 6,13,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $15$ $2$ $( 2,12)( 3,11)( 4,13)( 5, 9)( 6, 8)( 7,10)(14,15)$
$ 10, 2, 2, 1 $ $60$ $10$ $( 1,13)( 2,12, 5, 9, 8, 6,11, 3,14,15)( 4,10)$
$ 4, 4, 4, 1, 1, 1 $ $25$ $4$ $( 3, 9, 6,15)( 4, 7,13,10)( 5, 8,14,11)$
$ 12, 3 $ $50$ $12$ $( 1,11,15,13, 5, 3, 4, 2,12, 7, 8, 9)( 6,10,14)$
$ 4, 4, 4, 2, 1 $ $75$ $4$ $( 2,12)( 3,14, 6, 5)( 4, 7,13,10)( 8, 9,11,15)$
$ 4, 4, 4, 1, 1, 1 $ $25$ $4$ $( 3,15, 6, 9)( 4,10,13, 7)( 5,11,14, 8)$
$ 12, 3 $ $50$ $12$ $( 1,11, 9,13, 2,12, 7,14, 3,10, 8,15)( 4, 5, 6)$
$ 4, 4, 4, 2, 1 $ $75$ $4$ $( 2,12)( 3, 5, 6,14)( 4,10,13, 7)( 8,15,11, 9)$

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [600, 151]
Character table:   
      2  3  1  1  2  3   1  3  2  3   1  3   2  3  3   2  3
      3  1  .  .  1  .   .  1  1  .   .  1   1  .  1   1  .
      5  2  2  2  .  1   1  .  .  1   1  .   .  .  .   .  .

        1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4a 12a 4b 4c 12b 4d
     2P 1a 5a 5b 3a 1a  5b 1a 3a 1a  5a 2b  6a 2b 2b  6a 2b
     3P 1a 5a 5b 1a 2a 10a 2b 2b 2c 10b 4c  4c 4d 4a  4a 4b
     5P 1a 1a 1a 3a 2a  2a 2b 6a 2c  2c 4a 12a 4b 4c 12b 4d
     7P 1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4c 12b 4d 4a 12a 4b
    11P 1a 5a 5b 3a 2a 10a 2b 6a 2c 10b 4c 12b 4d 4a 12a 4b

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  1  1 -1  -1 -1 -1  1   1  A   A -A -A  -A  A
X.6      1  1  1  1 -1  -1 -1 -1  1   1 -A  -A  A  A   A -A
X.7      1  1  1  1  1   1 -1 -1 -1  -1  A   A  A -A  -A -A
X.8      1  1  1  1  1   1 -1 -1 -1  -1 -A  -A -A  A   A  A
X.9      2  2  2 -1  .   .  2 -1  .   . -2   1  . -2   1  .
X.10     2  2  2 -1  .   .  2 -1  .   .  2  -1  .  2  -1  .
X.11     2  2  2 -1  .   . -2  1  .   .  B  -A  . -B   A  .
X.12     2  2  2 -1  .   . -2  1  .   . -B   A  .  B  -A  .
X.13    12 -3  2  .  .   .  .  . -4   1  .   .  .  .   .  .
X.14    12 -3  2  .  .   .  .  .  4  -1  .   .  .  .   .  .
X.15    12  2 -3  . -4   1  .  .  .   .  .   .  .  .   .  .
X.16    12  2 -3  .  4  -1  .  .  .   .  .   .  .  .   .  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i