# Properties

 Label 40T25 Order $$80$$ n $$40$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5:OD_{16}$

## Group action invariants

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

## 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:  $C_4\times C_2$
16:  $C_8:C_2$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 8: $C_8:C_2$

Degree 10: $F_5$

Degree 20: 20T5

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

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [80, 33]
 Character table:  2 4 3 4 3 4 4 3 3 3 3 2 2 2 2 5 1 1 1 . . . . . . . 1 1 1 1 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 5a 10a 10b 10c 2P 1a 1a 1a 2b 2b 2b 4c 4b 4b 4c 5a 5a 5a 5a 3P 1a 2a 2b 4a 4c 4b 8c 8d 8a 8b 5a 10b 10a 10c 5P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 1a 2a 2a 2b 7P 1a 2a 2b 4a 4c 4b 8c 8d 8a 8b 5a 10b 10a 10c X.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 X.3 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 X.5 1 -1 1 1 -1 -1 B -B -B B 1 -1 -1 1 X.6 1 -1 1 1 -1 -1 -B B B -B 1 -1 -1 1 X.7 1 1 1 -1 -1 -1 B B -B -B 1 1 1 1 X.8 1 1 1 -1 -1 -1 -B -B B B 1 1 1 1 X.9 2 . -2 . A -A . . . . 2 . . -2 X.10 2 . -2 . -A A . . . . 2 . . -2 X.11 4 -4 4 . . . . . . . -1 1 1 -1 X.12 4 4 4 . . . . . . . -1 -1 -1 -1 X.13 4 . -4 . . . . . . . -1 C -C 1 X.14 4 . -4 . . . . . . . -1 -C C 1 A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(4) = -Sqrt(-1) = -i C = -E(5)+E(5)^2+E(5)^3-E(5)^4 = -Sqrt(5) = -r5