Properties

Label 15T30
Order \(750\)
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 $t$ :  $30$
CHM label :  $[5^{3}:2]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$
30:  $D_5\times C_3$
150:  $(C_5^2 : C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T30 x 7, 30T188 x 8

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 $ $6$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 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 $ $6$ $5$ $( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3, 6, 9,12,15)$
$ 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 $ $6$ $5$ $( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 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 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2,14,11, 8, 5)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
$ 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 $ $25$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 15 $ $50$ $15$ $( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)$
$ 15 $ $50$ $15$ $( 1,12, 2, 7, 3, 8,13, 9,14, 4,15, 5,10, 6,11)$
$ 6, 6, 3 $ $125$ $6$ $( 1,15,14,13, 3,11)( 2,10, 6, 8, 4,12)( 5, 7, 9)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $50$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 15 $ $50$ $15$ $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$
$ 6, 6, 3 $ $125$ $6$ $( 1, 8, 3, 4, 5, 6)( 2, 9,13,11,15, 7)(10,14,12)$

Group invariants

Order:  $750=2 \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [750, 30]
Character table: Data not available.