# Properties

 Label 40T8 Order $$40$$ n $$40$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:Q_8$

## Group action invariants

 Degree $n$ : $40$ Transitive number $t$ : $8$ Group : $C_5:Q_8$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,38,35,31,28,24,19,16,12,7,2,37,36,32,27,23,20,15,11,8)(3,40,33,30,26,21,18,14,10,5,4,39,34,29,25,22,17,13,9,6), (1,4,2,3)(5,37,6,38)(7,40,8,39)(9,35,10,36)(11,34,12,33)(13,31,14,32)(15,29,16,30)(17,28,18,27)(19,26,20,25)(21,23,22,24) $|\Aut(F/K)|$: $40$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$
10:  $D_{5}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8$

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

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

## Group invariants

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