# Properties

 Label 40T39 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 $t$: $39$ Group: $D_4\times D_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,4)(2,3)(5,40)(6,39)(7,37)(8,38)(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), (1,15,25,39,12,23,35,7,20,31,2,16,26,40,11,24,36,8,19,32)(3,13,28,38,10,22,34,5,18,29,4,14,27,37,9,21,33,6,17,30), (1,31)(2,32)(3,29)(4,30)(5,28)(6,27)(7,25)(8,26)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,37)(34,38)(35,39)(36,40)

## 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: $D_4\times C_2$

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 39]
 Character table:  2 4 3 4 3 4 4 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 10a 2f 20a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b 2P 1a 1a 1a 1a 1a 1a 2b 5a 1a 10c 5b 5b 5b 5b 10f 5a 5a 5a 1a 2b 3P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f 5b 10a 20a 10b 10c 5a 2g 4b 5P 1a 2a 2b 2c 2d 2e 4a 2g 2f 4b 2c 2b 1a 2g 4b 2c 2b 1a 2g 4b 7P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f 5b 10a 20a 10b 10c 5a 2g 4b 11P 1a 2a 2b 2c 2d 2e 4a 10a 2f 20a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b 13P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f 5b 10a 20a 10b 10c 5a 2g 4b 17P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f 5b 10a 20a 10b 10c 5a 2g 4b 19P 1a 2a 2b 2c 2d 2e 4a 10a 2f 20a 10b 10c 5a 10d 20b 10e 10f 5b 2g 4b 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