# Properties

 Label 40T18 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{10}.D_4$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $18$ Group: $C_{10}.D_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $20$ Generators: (1,22,4,24)(2,21,3,23)(5,18,7,20)(6,17,8,19)(9,15,12,13)(10,16,11,14)(25,40,27,38)(26,39,28,37)(29,36,31,33)(30,35,32,34), (1,18,36,9,25)(2,17,35,10,26)(3,19,34,11,28)(4,20,33,12,27)(5,23,38,14,29,6,24,37,13,30)(7,21,40,16,31,8,22,39,15,32)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$10$:  $D_{5}$
$16$:  $C_2^2:C_4$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $C_2^2:C_4$

Degree 10: $D_5$

Degree 20: 20T2, 20T11 x 2

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 19]
 Character table: not available.