Properties

Label 15T34
Order \(810\)
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 $t$ :  $34$
CHM label :  $[3^{4}]D(5)$
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)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T34 x 3, 15T35 x 4, 30T191 x 4, 30T192 x 4, 45T121 x 8, 45T122 x 8, 45T123, 45T124

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $5$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $5$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 5, 5, 5 $ $162$ $5$ $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 5, 5, 5 $ $162$ $5$ $( 1,12, 3, 4,10)( 2, 8, 9,15, 6)( 5,11, 7,13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1, 6,11)( 2,15)( 3,14,13, 9, 8, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$
$ 6, 6, 3 $ $45$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15,12,10, 7, 5)( 3, 9, 8,14,13, 4)$
$ 6, 6, 3 $ $45$ $6$ $( 1,11, 6)( 2,15,12,10, 7, 5)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $45$ $6$ $( 1, 6,11)( 2,15,12,10, 7, 5)( 3, 4)( 8, 9)(13,14)$

Group invariants

Order:  $810=2 \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [810, 101]
Character table: Data not available.