Properties

Label 15T42
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 $t$ :  $42$
CHM label :  $1/2[3^{4}:2]F(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,6,11)(4,14,9), (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:  $C_2$
4:  $C_4$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $F_5$

Low degree siblings

15T41, 30T295, 30T296, 30T298, 30T299, 30T302, 45T204, 45T210

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 $ $20$ $3$ $( 1,11, 6)( 4, 9,14)$
$ 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 $ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4, 9,14)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $324$ $5$ $( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 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, 3, 2, 2, 2 $ $90$ $6$ $( 1, 6,11)( 2,15)( 3,14,13, 9, 8, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 6, 6, 1, 1, 1 $ $45$ $6$ $( 2,15,12,10, 7, 5)( 3, 9, 8,14,13, 4)$
$ 12, 2, 1 $ $135$ $12$ $( 2, 3,15,14, 7,13, 5, 9,12, 8,10, 4)( 6,11)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 1,11)( 2, 3,15, 4)( 5,14, 7,13)( 8,10, 9,12)$
$ 12, 2, 1 $ $135$ $12$ $( 1, 6)( 2, 3,15, 9,12, 8,10,14, 7,13, 5, 4)$
$ 12, 2, 1 $ $135$ $12$ $( 2, 4,15,13, 7,14, 5, 8,12, 9,10, 3)( 6,11)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 1,11)( 2, 9,10, 3)( 4,15,13, 7)( 5, 8,12,14)$
$ 12, 2, 1 $ $135$ $12$ $( 1, 6)( 2,14, 5, 8,12, 4,15,13, 7, 9,10, 3)$

Group invariants

Order:  $1620=2^{2} \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1620, 421]
Character table: Data not available.