# Properties

 Label 40T14 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times F_5$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $14$ Group: $C_2\times F_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $40$ Generators: (1,21,3,23)(2,22,4,24)(5,11,40,33)(6,12,39,34)(7,10,37,36)(8,9,38,35)(13,25,29,20)(14,26,30,19)(15,28,31,18)(16,27,32,17), (1,30,33,8)(2,29,34,7)(3,31,36,6)(4,32,35,5)(9,13,28,23)(10,14,27,24)(11,15,25,22)(12,16,26,21)(17,39,20,38)(18,40,19,37)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

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

Degree 5: $F_5$

Degree 8: $C_4\times C_2$

Degree 10: $F_5$, $F_{5}\times C_2$ x 2

Degree 20: 20T5, 20T9, 20T13

## Low degree siblings

10T5 x 2

Siblings are shown with degree $\leq 10$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 12]
 Character table:  2 3 3 3 3 3 3 3 3 1 1 5 1 1 . . . . . . 1 1 1a 2a 2b 2c 4a 4b 4c 4d 5a 10a 2P 1a 1a 1a 1a 2b 2b 2b 2b 5a 5a 3P 1a 2a 2b 2c 4c 4d 4a 4b 5a 10a 5P 1a 2a 2b 2c 4a 4b 4c 4d 1a 2a 7P 1a 2a 2b 2c 4c 4d 4a 4b 5a 10a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 1 -1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 X.5 1 -1 -1 1 A -A -A A 1 -1 X.6 1 -1 -1 1 -A A A -A 1 -1 X.7 1 1 -1 -1 A A -A -A 1 1 X.8 1 1 -1 -1 -A -A A A 1 1 X.9 4 -4 . . . . . . -1 1 X.10 4 4 . . . . . . -1 -1 A = -E(4) = -Sqrt(-1) = -i