↑

Properties

Label 15T88
Order \(233280\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $88$
CHM label :  $[1/2.S(3)^{5}]A(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15), (1,13)(2,14)(3,6)(4,7)(8,11)(9,12), (1,11)(4,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
60:  $A_5$
960:  $C_2^4 : A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $A_5$

Low degree siblings

30T1643, 30T1647, 45T1278

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 $ $40$ $3$ $( 3, 8,13)( 5,10,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $80$ $3$ $( 1, 6,11)( 3, 8,13)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $80$ $3$ $( 1, 6,11)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,15,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $90$ $2$ $( 8,13)( 9,14)$
$ 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $540$ $6$ $( 5,10,15)( 8,13)( 9,14)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1 $ $1080$ $6$ $( 1, 6,11)( 5,10,15)( 8,13)( 9,14)$
$ 3, 3, 3, 2, 2, 1, 1 $ $720$ $6$ $( 1, 6,11)( 2, 7,12)( 5,10,15)( 8,13)( 9,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $405$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)$
$ 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $810$ $6$ $( 5,10,15)( 6,11)( 7,12)( 8,13)( 9,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $540$ $2$ $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)$
$ 3, 2, 2, 2, 2, 2, 2 $ $540$ $6$ $( 1, 2)( 3, 4)( 5,10,15)( 6, 7)( 8, 9)(11,12)(13,14)$
$ 3, 2, 2, 2, 2, 2, 2 $ $540$ $6$ $( 1, 2)( 3, 4)( 5,15,10)( 6, 7)( 8, 9)(11,12)(13,14)$
$ 6, 2, 2, 2, 1, 1, 1 $ $2160$ $6$ $( 1, 2)( 3, 4, 8, 9,13,14)( 6, 7)(11,12)$
$ 6, 3, 2, 2, 2 $ $2160$ $6$ $( 1, 2)( 3, 4, 8, 9,13,14)( 5,10,15)( 6, 7)(11,12)$
$ 6, 3, 2, 2, 2 $ $2160$ $6$ $( 1, 2)( 3, 4, 8, 9,13,14)( 5,15,10)( 6, 7)(11,12)$
$ 6, 6, 1, 1, 1 $ $2160$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)$
$ 6, 6, 3 $ $2160$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,10,15)$
$ 6, 6, 3 $ $2160$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$
$ 4, 2, 2, 2, 2, 2, 1 $ $9720$ $4$ $( 1, 2)( 3, 4)( 6, 7)( 8,14,13, 9)(10,15)(11,12)$
$ 6, 4, 2, 2, 1 $ $19440$ $12$ $( 1, 2, 6, 7,11,12)( 3, 4)( 8,14,13, 9)(10,15)$
$ 4, 4, 2, 2, 1, 1, 1 $ $4860$ $4$ $( 1, 2)( 3, 4)( 6,12,11, 7)( 8,14,13, 9)$
$ 4, 4, 3, 2, 2 $ $4860$ $12$ $( 1, 2)( 3, 4)( 5,10,15)( 6,12,11, 7)( 8,14,13, 9)$
$ 4, 4, 3, 2, 2 $ $4860$ $12$ $( 1, 2)( 3, 4)( 5,15,10)( 6,12,11, 7)( 8,14,13, 9)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $720$ $3$ $( 1, 2, 3)( 6, 7, 8)(11,12,13)$
$ 3, 3, 3, 3, 1, 1, 1 $ $1440$ $3$ $( 1, 2, 3)( 5,10,15)( 6, 7, 8)(11,12,13)$
$ 9, 1, 1, 1, 1, 1, 1 $ $1440$ $9$ $( 1, 2, 8, 6, 7,13,11,12, 3)$
$ 9, 3, 1, 1, 1 $ $2880$ $9$ $( 1, 2, 8, 6, 7,13,11,12, 3)( 5,10,15)$
$ 3, 3, 3, 3, 1, 1, 1 $ $1440$ $3$ $( 1, 2, 3)( 4, 9,14)( 6, 7, 8)(11,12,13)$
$ 3, 3, 3, 3, 3 $ $2880$ $3$ $( 1, 2, 3)( 4, 9,14)( 5,10,15)( 6, 7, 8)(11,12,13)$
$ 9, 3, 1, 1, 1 $ $2880$ $9$ $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)$
$ 9, 3, 3 $ $2880$ $9$ $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)( 5,10,15)$
$ 9, 3, 3 $ $2880$ $9$ $( 1, 2, 8, 6, 7,13,11,12, 3)( 4, 9,14)( 5,15,10)$
$ 6, 3, 2, 1, 1, 1, 1 $ $6480$ $6$ $( 1, 2, 3)( 6, 7,13,11,12, 8)( 9,14)$
$ 6, 3, 3, 2, 1 $ $12960$ $6$ $( 1, 2, 3)( 5,10,15)( 6, 7,13,11,12, 8)( 9,14)$
$ 3, 3, 3, 2, 2, 1, 1 $ $6480$ $6$ $( 1, 2, 3)( 6, 7, 8)( 9,14)(10,15)(11,12,13)$
$ 9, 2, 2, 1, 1 $ $12960$ $18$ $( 1, 2, 8, 6, 7,13,11,12, 3)( 9,14)(10,15)$
$ 6, 3, 2, 1, 1, 1, 1 $ $6480$ $6$ $( 1, 2, 3)( 6, 7,13,11,12, 8)(10,15)$
$ 6, 3, 3, 2, 1 $ $12960$ $6$ $( 1, 2, 3)( 4, 9,14)( 6, 7,13,11,12, 8)(10,15)$
$ 5, 5, 5 $ $15552$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)$
$ 15 $ $15552$ $15$ $( 1, 2, 3, 4,10, 6, 7, 8, 9,15,11,12,13,14, 5)$
$ 15 $ $15552$ $15$ $( 1, 2, 3, 4,15,11,12,13,14,10, 6, 7, 8, 9, 5)$
$ 5, 5, 5 $ $15552$ $5$ $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)(11,12,13,15,14)$
$ 15 $ $15552$ $15$ $( 1, 2, 3,10, 9, 6, 7, 8,15,14,11,12,13, 5, 4)$
$ 15 $ $15552$ $15$ $( 1, 2, 3,15,14,11,12,13,10, 9, 6, 7, 8, 5, 4)$

Group invariants

Order:  $233280=2^{6} \cdot 3^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.