# Properties

 Label 40T46 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{40}$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $46$ Group: $D_{40}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,22)(2,21)(3,24)(4,23)(5,19)(6,20)(7,18)(8,17)(9,15)(10,16)(11,13)(12,14)(25,38)(26,37)(27,40)(28,39)(29,35)(30,36)(31,34)(32,33), (1,8,10,14,20,23,27,30,35,39,3,5,11,15,18,21,25,31,34,38,2,7,9,13,19,24,28,29,36,40,4,6,12,16,17,22,26,32,33,37)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$16$:  $D_{8}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

Degree 10: $D_{10}$

Degree 20: $D_{20}$

## Low degree siblings

There are no siblings with degree $\leq 10$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $20$ $2$ $( 3, 4)( 5,40)( 6,39)( 7,38)( 8,37)( 9,34)(10,33)(11,36)(12,35)(13,31)(14,32) (15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,32,30,31)(33,35,34,36)(37,39,38,40)$ $40$ $2$ $40$ $( 1, 5, 9,16,20,21,28,32,35,38, 4, 8,11,13,17,23,25,29,33,39, 2, 6,10,15,19, 22,27,31,36,37, 3, 7,12,14,18,24,26,30,34,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $20$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,40)(10,39)(11,37)(12,38)(13,36)(14,35)(15,33) (16,34)(17,31)(18,32)(19,29)(20,30)(21,26)(22,25)(23,27)(24,28)$ $40$ $2$ $40$ $( 1, 6, 9,15,20,22,28,31,35,37, 4, 7,11,14,17,24,25,30,33,40, 2, 5,10,16,19, 21,27,32,36,38, 3, 8,12,13,18,23,26,29,34,39)$ $40$ $2$ $40$ $( 1, 7,10,13,20,24,27,29,35,40, 3, 6,11,16,18,22,25,32,34,37, 2, 8, 9,14,19, 23,28,30,36,39, 4, 5,12,15,17,21,26,31,33,38)$ $40$ $2$ $40$ $( 1, 8,10,14,20,23,27,30,35,39, 3, 5,11,15,18,21,25,31,34,38, 2, 7, 9,13,19, 24,28,29,36,40, 4, 6,12,16,17,22,26,32,33,37)$ $20, 20$ $2$ $20$ $( 1, 9,20,28,35, 4,11,17,25,33, 2,10,19,27,36, 3,12,18,26,34)( 5,16,21,32,38, 8,13,23,29,39, 6,15,22,31,37, 7,14,24,30,40)$ $20, 20$ $2$ $20$ $( 1,10,20,27,35, 3,11,18,25,34, 2, 9,19,28,36, 4,12,17,26,33)( 5,15,21,31,38, 7,13,24,29,40, 6,16,22,32,37, 8,14,23,30,39)$ $10, 10, 10, 10$ $2$ $10$ $( 1,11,19,26,35, 2,12,20,25,36)( 3, 9,17,27,34, 4,10,18,28,33)( 5,13,22,30,38, 6,14,21,29,37)( 7,16,23,31,40, 8,15,24,32,39)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,12,19,25,35)( 2,11,20,26,36)( 3,10,17,28,34)( 4, 9,18,27,33) ( 5,14,22,29,38)( 6,13,21,30,37)( 7,15,23,32,40)( 8,16,24,31,39)$ $40$ $2$ $40$ $( 1,13,27,40,11,22,34, 8,19,30, 4,15,26,38,10,24,35, 6,18,32, 2,14,28,39,12, 21,33, 7,20,29, 3,16,25,37, 9,23,36, 5,17,31)$ $40$ $2$ $40$ $( 1,14,27,39,11,21,34, 7,19,29, 4,16,26,37,10,23,35, 5,18,31, 2,13,28,40,12, 22,33, 8,20,30, 3,15,25,38, 9,24,36, 6,17,32)$ $40$ $2$ $40$ $( 1,15,28,37,11,24,33, 5,19,32, 3,13,26,39, 9,22,35, 7,17,30, 2,16,27,38,12, 23,34, 6,20,31, 4,14,25,40,10,21,36, 8,18,29)$ $40$ $2$ $40$ $( 1,16,28,38,11,23,33, 6,19,31, 3,14,26,40, 9,21,35, 8,17,29, 2,15,27,37,12, 24,34, 5,20,32, 4,13,25,39,10,22,36, 7,18,30)$ $20, 20$ $2$ $20$ $( 1,17,36, 9,25, 3,20,33,12,28, 2,18,35,10,26, 4,19,34,11,27)( 5,23,37,16,29, 7,21,39,14,32, 6,24,38,15,30, 8,22,40,13,31)$ $20, 20$ $2$ $20$ $( 1,18,36,10,25, 4,20,34,12,27, 2,17,35, 9,26, 3,19,33,11,28)( 5,24,37,15,29, 8,21,40,14,31, 6,23,38,16,30, 7,22,39,13,32)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,19,35,12,25)( 2,20,36,11,26)( 3,17,34,10,28)( 4,18,33, 9,27) ( 5,22,38,14,29)( 6,21,37,13,30)( 7,23,40,15,32)( 8,24,39,16,31)$ $10, 10, 10, 10$ $2$ $10$ $( 1,20,35,11,25, 2,19,36,12,26)( 3,18,34, 9,28, 4,17,33,10,27)( 5,21,38,13,29, 6,22,37,14,30)( 7,24,40,16,32, 8,23,39,15,31)$ $8, 8, 8, 8, 8$ $2$ $8$ $( 1,21, 4,23, 2,22, 3,24)( 5,28, 8,25, 6,27, 7,26)( 9,32,11,29,10,31,12,30) (13,33,15,36,14,34,16,35)(17,39,19,37,18,40,20,38)$ $8, 8, 8, 8, 8$ $2$ $8$ $( 1,22, 4,24, 2,21, 3,23)( 5,27, 8,26, 6,28, 7,25)( 9,31,11,30,10,32,12,29) (13,34,15,35,14,33,16,36)(17,40,19,38,18,39,20,37)$

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 7]
 Character table: not available.