# Properties

 Label 40T40 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_4\times D_5$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $40$ Group: $D_4\times D_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $8$ Generators: (1,32)(2,31)(3,30)(4,29)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,38)(34,37)(35,40)(36,39), (1,35)(2,36)(3,34)(4,33)(5,30)(6,29)(7,32)(8,31)(9,28)(10,27)(11,25)(12,26)(13,21)(14,22)(15,24)(16,23), (1,27,12,34,19,3,26,10,35,17)(2,28,11,33,20,4,25,9,36,18)(5,31,14,39,22,8,30,15,37,24)(6,32,13,40,21,7,29,16,38,23)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3

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: $D_{10}$ x 3

Degree 20: 20T8, 20T21 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 5,37)( 6,38)( 7,40)( 8,39)( 9,33)(10,34)(11,36)(12,35)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,26)(20,25)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 2)( 3, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,30) (14,29)(15,32)(16,31)(17,28)(18,27)(19,25)(20,26)(21,22)(23,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,38)( 8,37)( 9,36)(10,35)(11,33)(12,34)(13,32) (14,31)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,38,12,37)(13,35,14,36)(15,34,16,33) (17,32,18,31)(19,30,20,29)(21,26,22,25)(23,28,24,27)$ $20, 20$ $4$ $20$ $( 1, 5,11,13,19,22,25,29,35,37, 2, 6,12,14,20,21,26,30,36,38)( 3, 7, 9,15,17, 23,28,31,34,40, 4, 8,10,16,18,24,27,32,33,39)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,38)(10,37)(11,39)(12,40)(13,33)(14,34)(15,36) (16,35)(17,30)(18,29)(19,32)(20,31)(21,28)(22,27)(23,26)(24,25)$ $10, 10, 10, 10$ $4$ $10$ $( 1, 7,12,16,19,23,26,32,35,40)( 2, 8,11,15,20,24,25,31,36,39)( 3, 5,10,14,17, 22,27,30,34,37)( 4, 6, 9,13,18,21,28,29,33,38)$ $10, 10, 10, 10$ $4$ $10$ $( 1, 9,19,28,35, 4,12,18,26,33)( 2,10,20,27,36, 3,11,17,25,34)( 5,16,22,32,37, 7,14,23,30,40)( 6,15,21,31,38, 8,13,24,29,39)$ $10, 10, 10, 10$ $2$ $10$ $( 1,11,19,25,35, 2,12,20,26,36)( 3, 9,17,28,34, 4,10,18,27,33)( 5,13,22,29,37, 6,14,21,30,38)( 7,15,23,31,40, 8,16,24,32,39)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,12,19,26,35)( 2,11,20,25,36)( 3,10,17,27,34)( 4, 9,18,28,33) ( 5,14,22,30,37)( 6,13,21,29,38)( 7,16,23,32,40)( 8,15,24,31,39)$ $20, 20$ $4$ $20$ $( 1,13,25,37,12,21,36, 5,19,29, 2,14,26,38,11,22,35, 6,20,30)( 3,15,28,40,10, 24,33, 7,17,31, 4,16,27,39, 9,23,34, 8,18,32)$ $10, 10, 10, 10$ $4$ $10$ $( 1,15,26,39,12,24,35, 8,19,31)( 2,16,25,40,11,23,36, 7,20,32)( 3,13,27,38,10, 21,34, 6,17,29)( 4,14,28,37, 9,22,33, 5,18,30)$ $10, 10, 10, 10$ $4$ $10$ $( 1,17,35,10,26, 3,19,34,12,27)( 2,18,36, 9,25, 4,20,33,11,28)( 5,24,37,15,30, 8,22,39,14,31)( 6,23,38,16,29, 7,21,40,13,32)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,19,35,12,26)( 2,20,36,11,25)( 3,17,34,10,27)( 4,18,33, 9,28) ( 5,22,37,14,30)( 6,21,38,13,29)( 7,23,40,16,32)( 8,24,39,15,31)$ $10, 10, 10, 10$ $2$ $10$ $( 1,20,35,11,26, 2,19,36,12,25)( 3,18,34, 9,27, 4,17,33,10,28)( 5,21,37,13,30, 6,22,38,14,29)( 7,24,40,15,32, 8,23,39,16,31)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,21, 2,22)( 3,24, 4,23)( 5,26, 6,25)( 7,27, 8,28)( 9,32,10,31)(11,30,12,29) (13,36,14,35)(15,33,16,34)(17,39,18,40)(19,38,20,37)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,27)( 6,28)( 7,26)( 8,25)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,36)(16,35)(17,37)(18,38)(19,40)(20,39)$

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 39]
 Character table:  2 4 4 4 4 3 3 3 2 3 2 2 3 3 2 2 2 3 3 3 3 5 1 . 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 2e 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g 2P 1a 1a 1a 1a 1a 1a 2b 10c 1a 5a 5b 5b 5b 10f 5b 5a 5a 5a 2b 1a 3P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g 5P 1a 2a 2b 2c 2d 2e 4a 4b 2f 2g 2d 2b 1a 4b 2g 2d 1a 2b 4b 2g 7P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g 11P 1a 2a 2b 2c 2d 2e 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g 13P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g 17P 1a 2a 2b 2c 2d 2e 4a 20b 2f 10d 10e 10f 5b 20a 10a 10b 5a 10c 4b 2g 19P 1a 2a 2b 2c 2d 2e 4a 20a 2f 10a 10b 10c 5a 20b 10d 10e 5b 10f 4b 2g X.1 1 1 1 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 1 1 1 -1 X.3 1 -1 1 -1 -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 1 1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 X.6 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 X.7 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 X.8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 X.9 2 -2 -2 2 . . . . . . . -2 2 . . . 2 -2 . . X.10 2 2 -2 -2 . . . . . . . -2 2 . . . 2 -2 . . X.11 2 . 2 . -2 . . A . -A *A -*A -*A *A -*A A -A -A -2 2 X.12 2 . 2 . -2 . . *A . -*A A -A -A A -A *A -*A -*A -2 2 X.13 2 . 2 . -2 . . -*A . *A A -A -A -A A *A -*A -*A 2 -2 X.14 2 . 2 . -2 . . -A . A *A -*A -*A -*A *A A -A -A 2 -2 X.15 2 . 2 . 2 . . A . A -*A -*A -*A *A *A -A -A -A -2 -2 X.16 2 . 2 . 2 . . *A . *A -A -A -A A A -*A -*A -*A -2 -2 X.17 2 . 2 . 2 . . -*A . -*A -A -A -A -A -A -*A -*A -*A 2 2 X.18 2 . 2 . 2 . . -A . -A -*A -*A -*A -*A -*A -A -A -A 2 2 X.19 4 . -4 . . . . . . . . B -B . . . -*B *B . . X.20 4 . -4 . . . . . . . . *B -*B . . . -B B . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -2*E(5)^2-2*E(5)^3 = 1+Sqrt(5) = 1+r5