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Properties

Label 15T52
Order \(3240\)
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 $t$ :  $52$
CHM label :  $[3^{4}:2]F(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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), (1,7,4,13)(2,14,8,11)(3,6,12,9)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $F_5$

Low degree siblings

15T52, 30T447 x 2, 30T448 x 2, 30T449, 30T452 x 2, 30T453 x 2, 45T309, 45T313

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 $ $20$ $3$ $( 1, 6,11)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 2, 7,12)( 3,13, 8)( 4,14, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1,11, 6)( 2, 7,12)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $81$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$
$ 5, 5, 5 $ $324$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 10, 5 $ $324$ $10$ $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $45$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 6, 3, 2, 2, 2 $ $180$ $6$ $( 1, 6,11)( 2, 5)( 3, 4,13,14, 8, 9)( 7,10)(12,15)$
$ 6, 6, 1, 1, 1 $ $90$ $6$ $( 2, 5, 7,10,12,15)( 3, 9,13, 4, 8,14)$
$ 6, 6, 3 $ $90$ $6$ $( 1, 6,11)( 2, 5, 7,10,12,15)( 3, 4, 8, 9,13,14)$
$ 6, 2, 2, 2, 2, 1 $ $180$ $6$ $( 2, 5)( 3,14,13, 4, 8, 9)( 6,11)( 7,15)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 1 $ $45$ $2$ $( 1, 6)( 2, 5)( 3, 9)( 4, 8)( 7,15)(10,12)(13,14)$
$ 6, 6, 2, 1 $ $180$ $6$ $( 2, 5, 7,15,12,10)( 3,14, 8, 9,13, 4)( 6,11)$
$ 4, 4, 4, 1, 1, 1 $ $135$ $4$ $( 2, 8, 5,14)( 3,15, 9,12)( 4, 7,13,10)$
$ 12, 3 $ $270$ $12$ $( 1, 6,11)( 2, 8, 5, 9,12, 3,15, 4, 7,13,10,14)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 2,13,15,14)( 3,10, 4,12)( 5, 9, 7, 8)( 6,11)$
$ 12, 2, 1 $ $270$ $12$ $( 1, 6)( 2,13,15, 9, 7, 8, 5, 4,12, 3,10,14)$
$ 4, 4, 4, 1, 1, 1 $ $135$ $4$ $( 2,14, 5, 8)( 3,12, 9,15)( 4,10,13, 7)$
$ 12, 3 $ $270$ $12$ $( 1, 6,11)( 2, 9,15, 3,12, 4,10,13, 7,14, 5, 8)$
$ 4, 4, 4, 2, 1 $ $135$ $4$ $( 2, 9,10, 8)( 3, 7, 4,15)( 5,13,12,14)( 6,11)$
$ 12, 2, 1 $ $270$ $12$ $( 1, 6)( 2, 4,15, 3, 7,14, 5,13,12, 9,10, 8)$

Group invariants

Order:  $3240=2^{3} \cdot 3^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.