# Properties

 Label 40T32 Order $$80$$ n $$40$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:D_8$

## Group action invariants

 Degree $n$ : $40$ Transitive number $t$ : $32$ Group : $C_5:D_8$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (3,4)(5,40)(6,39)(7,37)(8,38)(9,33)(10,34)(11,36)(12,35)(13,32)(14,31)(15,29)(16,30)(17,27)(18,28)(19,26)(20,25)(21,24)(22,23), (1,38,4,39,2,37,3,40)(5,34,7,35,6,33,8,36)(9,32,12,29,10,31,11,30)(13,27,16,26,14,28,15,25)(17,23,20,22,18,24,19,21) $|\Aut(F/K)|$: $2$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

Degree 10: $D_{10}$

Degree 20: 20T7

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [80, 15]
 Character table:  2 4 2 4 3 2 3 3 2 2 3 3 2 2 2 3 3 2 5 1 . 1 1 1 . . 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 4a 10a 8a 8b 10b 20a 5a 10c 10d 10e 20b 5b 10f 2c 2P 1a 1a 1a 2b 5a 4a 4a 5a 10f 5b 5b 5b 5b 10c 5a 5a 1a 3P 1a 2a 2b 4a 10d 8b 8a 10e 20b 5b 10f 10b 10a 20a 5a 10c 2c 5P 1a 2a 2b 4a 2c 8b 8a 2c 4a 1a 2b 2c 2c 4a 1a 2b 2c 7P 1a 2a 2b 4a 10e 8a 8b 10d 20b 5b 10f 10a 10b 20a 5a 10c 2c 11P 1a 2a 2b 4a 10a 8b 8a 10b 20a 5a 10c 10d 10e 20b 5b 10f 2c 13P 1a 2a 2b 4a 10d 8b 8a 10e 20b 5b 10f 10b 10a 20a 5a 10c 2c 17P 1a 2a 2b 4a 10e 8a 8b 10d 20b 5b 10f 10a 10b 20a 5a 10c 2c 19P 1a 2a 2b 4a 10b 8b 8a 10a 20a 5a 10c 10e 10d 20b 5b 10f 2c X.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 X.3 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 X.5 2 . 2 -2 . . . . -2 2 2 . . -2 2 2 . X.6 2 . -2 . . D -D . . 2 -2 . . . 2 -2 . X.7 2 . -2 . . -D D . . 2 -2 . . . 2 -2 . X.8 2 . 2 -2 A . . -A *C -*C -*C -B B C -C -C . X.9 2 . 2 -2 B . . -B C -C -C A -A *C -*C -*C . X.10 2 . 2 -2 -B . . B C -C -C -A A *C -*C -*C . X.11 2 . 2 -2 -A . . A *C -*C -*C B -B C -C -C . X.12 2 . 2 2 C . . C -*C -*C -*C *C *C -C -C -C -2 X.13 2 . 2 2 *C . . *C -C -C -C C C -*C -*C -*C -2 X.14 2 . 2 2 -*C . . -*C -C -C -C -C -C -*C -*C -*C 2 X.15 2 . 2 2 -C . . -C -*C -*C -*C -*C -*C -C -C -C 2 X.16 4 . -4 . . . . . . E -E . . . *E -*E . X.17 4 . -4 . . . . . . *E -*E . . . E -E . A = -E(5)+E(5)^4 B = -E(5)^2+E(5)^3 C = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 D = -E(8)+E(8)^3 = -Sqrt(2) = -r2 E = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5