# Properties

 Label 40T4 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times C_5:C_4$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $4$ Group: $C_2\times C_5:C_4$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $40$ Generators: (1,11,18,28,36,3,10,19,25,34)(2,12,17,27,35,4,9,20,26,33)(5,16,24,32,38,7,13,21,29,39)(6,15,23,31,37,8,14,22,30,40), (1,38,4,40)(2,37,3,39)(5,33,8,36)(6,34,7,35)(9,30,11,32)(10,29,12,31)(13,27,15,25)(14,28,16,26)(17,23,19,21)(18,24,20,22)

## Low degree resolvents

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

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

Degree 20: 20T2 x 2, 20T4

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

## Group invariants

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