# Properties

 Label 40T12 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{20}$

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $D_4$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4, $D_{20}$ 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,35)(10,36)(11,34)(12,33)(13,31) (14,32)(15,30)(16,29)(17,27)(18,28)(19,26)(20,25)(21,24)(22,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,37)(10,38)(11,39)(12,40)(13,35)(14,36)(15,34) (16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$ $20, 20$ $2$ $20$ $( 1, 7,12,15,20,23,28,32,35,38, 2, 8,11,16,19,24,27,31,36,37)( 3, 6,10,13,17, 21,26,29,34,40, 4, 5, 9,14,18,22,25,30,33,39)$ $20, 20$ $2$ $20$ $( 1, 8,12,16,20,24,28,31,35,37, 2, 7,11,15,19,23,27,32,36,38)( 3, 5,10,14,17, 22,26,30,34,39, 4, 6, 9,13,18,21,25,29,33,40)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,11,20,27,35)( 2,12,19,28,36)( 3, 9,17,25,34)( 4,10,18,26,33) ( 5,13,22,29,39)( 6,14,21,30,40)( 7,16,23,31,38)( 8,15,24,32,37)$ $10, 10, 10, 10$ $2$ $10$ $( 1,12,20,28,35, 2,11,19,27,36)( 3,10,17,26,34, 4, 9,18,25,33)( 5,14,22,30,39, 6,13,21,29,40)( 7,15,23,32,38, 8,16,24,31,37)$ $20, 20$ $2$ $20$ $( 1,15,28,38,11,24,36, 7,20,32, 2,16,27,37,12,23,35, 8,19,31)( 3,13,26,40, 9, 22,33, 6,17,29, 4,14,25,39,10,21,34, 5,18,30)$ $20, 20$ $2$ $20$ $( 1,16,28,37,11,23,36, 8,20,31, 2,15,27,38,12,24,35, 7,19,32)( 3,14,26,39, 9, 21,33, 5,17,30, 4,13,25,40,10,22,34, 6,18,29)$ $10, 10, 10, 10$ $2$ $10$ $( 1,19,35,12,27, 2,20,36,11,28)( 3,18,34,10,25, 4,17,33, 9,26)( 5,21,39,14,29, 6,22,40,13,30)( 7,24,38,15,31, 8,23,37,16,32)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,20,35,11,27)( 2,19,36,12,28)( 3,17,34, 9,25)( 4,18,33,10,26) ( 5,22,39,13,29)( 6,21,40,14,30)( 7,23,38,16,31)( 8,24,37,15,32)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,23, 2,24)( 3,21, 4,22)( 5,25, 6,26)( 7,28, 8,27)( 9,30,10,29)(11,31,12,32) (13,34,14,33)(15,35,16,36)(17,40,18,39)(19,37,20,38)$

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 6]
 Character table:  2 3 3 2 2 2 2 2 2 2 2 2 2 2 5 1 1 . . 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 20a 20b 5a 10a 20c 20d 10b 5b 4a 2P 1a 1a 1a 1a 10a 10a 5b 5b 10b 10b 5a 5a 2a 3P 1a 2a 2b 2c 20c 20d 5b 10b 20b 20a 10a 5a 4a 5P 1a 2a 2b 2c 4a 4a 1a 2a 4a 4a 2a 1a 4a 7P 1a 2a 2b 2c 20d 20c 5b 10b 20a 20b 10a 5a 4a 11P 1a 2a 2b 2c 20b 20a 5a 10a 20d 20c 10b 5b 4a 13P 1a 2a 2b 2c 20d 20c 5b 10b 20a 20b 10a 5a 4a 17P 1a 2a 2b 2c 20c 20d 5b 10b 20b 20a 10a 5a 4a 19P 1a 2a 2b 2c 20a 20b 5a 10a 20c 20d 10b 5b 4a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 X.3 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 X.4 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 X.5 2 -2 . . . . 2 -2 . . -2 2 . X.6 2 -2 . . A -A -*C *C -B B C -C . X.7 2 -2 . . B -B -C C A -A *C -*C . X.8 2 -2 . . -B B -C C -A A *C -*C . X.9 2 -2 . . -A A -*C *C B -B C -C . X.10 2 2 . . C C -*C -*C *C *C -C -C -2 X.11 2 2 . . *C *C -C -C C C -*C -*C -2 X.12 2 2 . . -*C -*C -C -C -C -C -*C -*C 2 X.13 2 2 . . -C -C -*C -*C -*C -*C -C -C 2 A = -E(20)+E(20)^9 B = -E(20)^13+E(20)^17 C = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5