# Properties

 Label 40T29 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times C_4\times D_5$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $29$ Group: $C_2\times C_4\times D_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $8$ Generators: (1,4)(2,3)(5,39)(6,40)(7,37)(8,38)(9,36)(10,35)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,26)(18,25)(19,28)(20,27)(21,24)(22,23), (1,18)(2,17)(3,20)(4,19)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(21,40)(22,39)(23,38)(24,37)(25,34)(26,33)(27,35)(28,36)(29,31)(30,32), (1,14,28,39,12,22,33,8,20,29,3,15,26,37,10,24,36,6,18,31)(2,13,27,40,11,21,34,7,19,30,4,16,25,38,9,23,35,5,17,32)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_4\times C_2$ x 6, $C_2^3$
$10$:  $D_{5}$
$20$:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 5: $D_{5}$

Degree 8: $C_4\times C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T6 x 2, 20T8

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 36]
 Character table: not available.