# Properties

 Label 40T3 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5:C_8$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $3$ Group: $C_5:C_8$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $40$ Generators: (1,27,10,33,17,3,26,11,35,19,2,28,9,34,18,4,25,12,36,20)(5,31,14,40,23,8,30,15,37,22,6,32,13,39,24,7,29,16,38,21), (1,5,3,8,2,6,4,7)(9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35)(17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 8: $C_8$

Degree 10: $D_5$

Degree 20: 20T2

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 1]
 Character table:  2 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 5 1 1 1 1 . . . . 1 1 1 1 1 1 1 1 1a 2a 4a 4b 8a 8b 8c 8d 5a 10a 20a 20b 5b 10b 20c 20d 2P 1a 1a 2a 2a 4a 4a 4b 4b 5b 5b 10b 10b 5a 5a 10a 10a 3P 1a 2a 4b 4a 8d 8c 8b 8a 5b 10b 20c 20d 5a 10a 20a 20b 5P 1a 2a 4a 4b 8b 8a 8d 8c 1a 2a 4b 4a 1a 2a 4a 4b 7P 1a 2a 4b 4a 8c 8d 8a 8b 5b 10b 20c 20d 5a 10a 20a 20b 11P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c 13P 1a 2a 4a 4b 8b 8a 8d 8c 5b 10b 20d 20c 5a 10a 20b 20a 17P 1a 2a 4a 4b 8a 8b 8c 8d 5b 10b 20d 20c 5a 10a 20b 20a 19P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 20b 20a 5b 10b 20d 20c X.1 1 1 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 1 1 1 X.3 1 -1 A -A C -C /C -/C 1 -1 -A A 1 -1 A -A X.4 1 -1 A -A -C C -/C /C 1 -1 -A A 1 -1 A -A X.5 1 -1 -A A -/C /C -C C 1 -1 A -A 1 -1 -A A X.6 1 -1 -A A /C -/C C -C 1 -1 A -A 1 -1 -A A X.7 1 1 -1 -1 A A -A -A 1 1 -1 -1 1 1 -1 -1 X.8 1 1 -1 -1 -A -A A A 1 1 -1 -1 1 1 -1 -1 X.9 2 -2 B -B . . . . D -D E -E *D -*D -F F X.10 2 -2 B -B . . . . *D -*D F -F D -D -E E X.11 2 -2 -B B . . . . D -D -E E *D -*D F -F X.12 2 -2 -B B . . . . *D -*D -F F D -D E -E X.13 2 2 -2 -2 . . . . D D -D -D *D *D -*D -*D X.14 2 2 -2 -2 . . . . *D *D -*D -*D D D -D -D X.15 2 2 2 2 . . . . D D D D *D *D *D *D X.16 2 2 2 2 . . . . *D *D *D *D D D D D A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i C = -E(8)^3 D = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 E = E(20)^13+E(20)^17 F = E(20)+E(20)^9