Properties

Label 15T26
Order \(405\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^4:C_5$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $26$
Group :  $C_3^4:C_5$
CHM label :  $[3^{4}]5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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)
5:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $C_5$

Low degree siblings

15T26 x 7, 45T61 x 2, 45T62 x 16, 45T63 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 $ $1$ $1$ $()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 4, 9,14)( 5,15,10)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 4,14, 9)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3,13, 8)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3, 8,13)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2, 7,12)( 3,13, 8)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $5$ $3$ $( 2,12, 7)( 3,13, 8)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)$
$ 5, 5, 5 $ $81$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)$
$ 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 $ $5$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$

Group invariants

Order:  $405=3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [405, 15]
Character table: Data not available.