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Properties

Label 15T77
Order \(29160\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $77$
CHM label :  $1/2[3^{5}:2]S(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14)(2,7)(4,11)(5,10)(6,9)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
120:  $S_5$
360:  $\GL(2,4):C_2$
9720:  15T62

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $S_5$

Low degree siblings

15T77 x 5, 30T1020 x 6, 30T1024 x 2, 45T697 x 2, 45T714 x 6, 45T717 x 3

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $3$ $( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 3, 8,13)( 5,10,15)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 6,11)( 3, 8,13)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $60$ $3$ $( 1, 6,11)( 3, 8,13)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $40$ $3$ $( 1, 6,11)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 1, 1, 1 $ $30$ $3$ $( 1, 6,11)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $10$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 3, 3, 3, 3, 3 $ $20$ $3$ $( 1, 6,11)( 2, 7,12)( 3,13, 8)( 4, 9,14)( 5,15,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $810$ $2$ $( 1,14)( 2, 7)( 4,11)( 5,10)( 6, 9)( 8,13)$
$ 6, 2, 2, 2, 1, 1, 1 $ $1620$ $6$ $( 1,14, 6, 9,11, 4)( 2, 7)( 5,10)( 8,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $135$ $2$ $( 1, 4)( 2, 5)( 6, 9)( 7,10)(11,14)(12,15)$
$ 6, 2, 2, 2, 1, 1, 1 $ $540$ $6$ $( 1, 4)( 2,10, 7,15,12, 5)( 6, 9)(11,14)$
$ 3, 2, 2, 2, 2, 2, 2 $ $270$ $6$ $( 1, 4)( 2, 5)( 3, 8,13)( 6, 9)( 7,10)(11,14)(12,15)$
$ 6, 3, 2, 2, 2 $ $540$ $6$ $( 1, 4)( 2,10, 7,15,12, 5)( 3, 8,13)( 6, 9)(11,14)$
$ 6, 3, 2, 2, 2 $ $540$ $6$ $( 1, 4)( 2,15,12,10, 7, 5)( 3, 8,13)( 6, 9)(11,14)$
$ 6, 6, 1, 1, 1 $ $270$ $6$ $( 1, 4, 6, 9,11,14)( 2,10, 7,15,12, 5)$
$ 6, 6, 1, 1, 1 $ $270$ $6$ $( 1, 4, 6, 9,11,14)( 2,15,12,10, 7, 5)$
$ 6, 6, 3 $ $270$ $6$ $( 1, 4, 6, 9,11,14)( 2,10, 7,15,12, 5)( 3, 8,13)$
$ 6, 6, 3 $ $540$ $6$ $( 1, 4, 6, 9,11,14)( 2,15,12,10, 7, 5)( 3, 8,13)$
$ 6, 6, 3 $ $270$ $6$ $( 1, 4, 6, 9,11,14)( 2,10, 7,15,12, 5)( 3,13, 8)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $180$ $3$ $( 1, 4, 7)( 2,11,14)( 6, 9,12)$
$ 3, 3, 3, 3, 1, 1, 1 $ $360$ $3$ $( 1, 4, 7)( 2,11,14)( 5,10,15)( 6, 9,12)$
$ 3, 3, 3, 3, 1, 1, 1 $ $360$ $3$ $( 1, 4, 7)( 2,11,14)( 5,15,10)( 6, 9,12)$
$ 3, 3, 3, 3, 3 $ $360$ $3$ $( 1, 4, 7)( 2,11,14)( 3, 8,13)( 5,10,15)( 6, 9,12)$
$ 3, 3, 3, 3, 3 $ $180$ $3$ $( 1, 4, 7)( 2,11,14)( 3, 8,13)( 5,15,10)( 6, 9,12)$
$ 3, 3, 3, 3, 3 $ $180$ $3$ $( 1, 4, 7)( 2,11,14)( 3,13, 8)( 5,10,15)( 6, 9,12)$
$ 9, 1, 1, 1, 1, 1, 1 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)$
$ 9, 3, 1, 1, 1 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 5,10,15)$
$ 9, 3, 1, 1, 1 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 5,15,10)$
$ 9, 3, 1, 1, 1 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3, 8,13)$
$ 9, 3, 3 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3, 8,13)( 5,10,15)$
$ 9, 3, 3 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3, 8,13)( 5,15,10)$
$ 9, 3, 1, 1, 1 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3,13, 8)$
$ 9, 3, 3 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3,13, 8)( 5,10,15)$
$ 9, 3, 3 $ $360$ $9$ $( 1, 4, 7, 6, 9,12,11,14, 2)( 3,13, 8)( 5,15,10)$
$ 6, 3, 2, 2, 2 $ $1620$ $6$ $( 1,14, 7,11, 4, 2)( 3,15)( 5,13)( 6, 9,12)( 8,10)$
$ 6, 6, 3 $ $1620$ $6$ $( 1,14, 7,11, 4, 2)( 3, 5,13,10, 8,15)( 6, 9,12)$
$ 6, 6, 3 $ $1620$ $6$ $( 1,14, 7,11, 4, 2)( 3,10, 8, 5,13,15)( 6, 9,12)$
$ 4, 4, 4, 2, 1 $ $2430$ $4$ $( 1,14, 7, 5)( 2,10,11, 4)( 6, 9,12,15)( 8,13)$
$ 12, 2, 1 $ $2430$ $12$ $( 1,14, 7,10,11, 4, 2,15, 6, 9,12, 5)( 8,13)$
$ 12, 2, 1 $ $2430$ $12$ $( 1,14, 7,15, 6, 9,12,10,11, 4, 2, 5)( 8,13)$
$ 5, 5, 5 $ $1944$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $1944$ $15$ $( 1, 4, 7,15, 3, 6, 9,12, 5, 8,11,14, 2,10,13)$
$ 15 $ $1944$ $15$ $( 1, 4, 7, 5, 8,11,14, 2,15, 3, 6, 9,12,10,13)$

Group invariants

Order:  $29160=2^{3} \cdot 3^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.