Properties

Label 15T43
Order \(1620\)
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 $t$ :  $43$
CHM label :  $[3^{4}:2]D(5)$
Parity:  $-1$
Primitive:  No
Generators:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(4,14,9), (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(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:  $V_4$
10:  $D_{5}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Low degree siblings

15T43 x 7, 30T290 x 8, 30T291 x 8, 30T297 x 8, 45T205 x 16, 45T206 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 $ $1$ $1$ $()$
$ 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 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 4, 9,14)( 5,10,15)$
$ 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 $ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 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 $ $90$ $6$ $( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $90$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 3, 2, 2, 2 $ $90$ $6$ $( 1,11, 6)( 2,15, 7, 5,12,10)( 3, 4)( 8, 9)(13,14)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $81$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 10, 5 $ $162$ $10$ $( 1, 4,12, 5,13, 6,14, 2,15, 3)( 7,10, 8,11, 9)$
$ 10, 5 $ $162$ $10$ $( 1,12,13, 9, 5,11, 2, 8,14,15)( 3, 4,10, 6, 7)$
$ 6, 2, 2, 2, 2, 1 $ $90$ $6$ $( 2,15,12, 5, 7,10)( 3, 4)( 6,11)( 8,14)( 9,13)$
$ 6, 6, 2, 1 $ $90$ $6$ $( 1,11)( 2,15,12, 5, 7,10)( 3, 9,13,14, 8, 4)$
$ 6, 6, 2, 1 $ $90$ $6$ $( 1, 6)( 2,15,12, 5, 7,10)( 3,14, 8, 9,13, 4)$
$ 6, 2, 2, 2, 2, 1 $ $90$ $6$ $( 2,15)( 3,14, 8, 9,13, 4)( 5,12)( 6,11)( 7,10)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 1,11)( 2,15)( 3, 4)( 5,12)( 7,10)( 8,14)( 9,13)$

Group invariants

Order:  $1620=2^{2} \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1620, 422]
Character table: Data not available.