Properties

Label 15T40
Order \(1500\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $40$
CHM label :  $[5^{3}:2]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,8)(3,12)(6,9)(7,13)(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), (3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
10:  $D_{5}$
12:  $D_{6}$
20:  $D_{10}$
60:  $D_5\times S_3$
300:  $((C_5^2 : C_3):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T40 x 3, 30T273 x 4, 30T275 x 4, 30T280 x 2, 30T284 x 4

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 6, 9,12,15)$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $5$ $( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $2$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $125$ $2$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $100$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 15 $ $100$ $15$ $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$
$ 6, 6, 3 $ $250$ $6$ $( 1, 8, 3, 4, 5, 6)( 2, 9,13,11,15, 7)(10,14,12)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1,11)( 2, 7)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3, 6, 9,12,15)( 4,14)( 5,10)( 8,13)$
$ 5, 2, 2, 2, 2, 2 $ $30$ $10$ $( 1,11)( 2, 7)( 3, 9,15, 6,12)( 4,14)( 5,10)( 8,13)$
$ 10, 1, 1, 1, 1, 1 $ $30$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)$
$ 10, 5 $ $30$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 6, 9,12,15)$
$ 10, 5 $ $30$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3, 9,15, 6,12)$
$ 10, 5 $ $30$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3,15,12, 9, 6)$
$ 10, 5 $ $30$ $10$ $( 1,14, 4, 2, 7, 5,10, 8,13,11)( 3,12, 6,15, 9)$
$ 10, 1, 1, 1, 1, 1 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3, 6, 9,12,15)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3, 9,15, 6,12)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,15,12, 9, 6)$
$ 10, 5 $ $30$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,12, 6,15, 9)$
$ 10, 2, 2, 1 $ $150$ $10$ $( 1, 8, 4, 5, 7, 2,10,14,13,11)( 6,15)( 9,12)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $75$ $2$ $( 1,11)( 2,10)( 4, 8)( 5, 7)( 6,15)( 9,12)(13,14)$
$ 10, 2, 2, 1 $ $150$ $10$ $( 1, 5, 7,14,13, 8, 4, 2,10,11)( 6,15)( 9,12)$

Group invariants

Order:  $1500=2^{2} \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1500, 125]
Character table: Data not available.