Properties

 Label 15T42 Order $$1620$$ n $$15$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

Related objects

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

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.