# Properties

 Label 40T26 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2:F_5$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $26$ Group: $C_2^2:F_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $20$ Generators: (1,30,33,8)(2,29,34,7)(3,31,36,5)(4,32,35,6)(9,14,28,23)(10,13,27,24)(11,16,26,22)(12,15,25,21)(17,39,20,38)(18,40,19,37), (1,11,19,27,36,2,12,20,28,35)(3,10,18,26,33,4,9,17,25,34)(5,15,23,30,37)(6,16,24,29,38)(7,13,22,32,39)(8,14,21,31,40)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 5: $F_5$

Degree 8: $C_2^2:C_4$

Degree 10: $F_5$

Degree 20: 20T5, 20T22 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,37)( 8,38)( 9,36)(10,35)(11,34)(12,33)(13,30) (14,29)(15,32)(16,31)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 3)( 2, 4)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,34)(12,33)(13,29) (14,30)(15,31)(16,32)(17,27)(18,28)(19,25)(20,26)(21,23)(22,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 4)( 2, 3)( 5,39)( 6,40)( 7,37)( 8,38)( 9,35)(10,36)(11,33)(12,34)(13,30) (14,29)(15,32)(16,31)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 5,17,13)( 2, 6,18,14)( 3, 8,20,16)( 4, 7,19,15)( 9,31,11,29)(10,32,12,30) (21,27,38,33)(22,28,37,34)(23,26,39,36)(24,25,40,35)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 5,18,14)( 2, 6,17,13)( 3, 8,19,15)( 4, 7,20,16)( 9,31,12,30)(10,32,11,29) (21,28,37,33)(22,27,38,34)(23,25,40,36)(24,26,39,35)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 7,33,29)( 2, 8,34,30)( 3, 6,36,32)( 4, 5,35,31)( 9,24,28,13)(10,23,27,14) (11,21,26,15)(12,22,25,16)(17,37,20,40)(18,38,19,39)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 7,34,30)( 2, 8,33,29)( 3, 6,35,31)( 4, 5,36,32)( 9,24,27,14)(10,23,28,13) (11,21,25,16)(12,22,26,15)(17,37,19,39)(18,38,20,40)$ $10, 10, 5, 5, 5, 5$ $4$ $10$ $( 1,11,19,27,36, 2,12,20,28,35)( 3,10,18,26,33, 4, 9,17,25,34)( 5,15,23,30,37) ( 6,16,24,29,38)( 7,13,22,32,39)( 8,14,21,31,40)$ $10, 10, 10, 10$ $4$ $10$ $( 1,11,19,27,36, 2,12,20,28,35)( 3,10,18,26,33, 4, 9,17,25,34)( 5,16,23,29,37, 6,15,24,30,38)( 7,14,22,31,39, 8,13,21,32,40)$ $5, 5, 5, 5, 5, 5, 5, 5$ $4$ $5$ $( 1,12,19,28,36)( 2,11,20,27,35)( 3, 9,18,25,33)( 4,10,17,26,34) ( 5,15,23,30,37)( 6,16,24,29,38)( 7,13,22,32,39)( 8,14,21,31,40)$ $10, 10, 5, 5, 5, 5$ $4$ $10$ $( 1,12,19,28,36)( 2,11,20,27,35)( 3, 9,18,25,33)( 4,10,17,26,34) ( 5,16,23,29,37, 6,15,24,30,38)( 7,14,22,31,39, 8,13,21,32,40)$

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 34]
 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 2c 2d 2e 4a 4b 4c 4d 10a 10b 5a 10c 2P 1a 1a 1a 1a 1a 1a 2e 2d 2d 2e 5a 5a 5a 5a 3P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10c 10b 5a 10a 5P 1a 2a 2b 2c 2d 2e 4a 4b 4c 4d 2a 2b 1a 2a 7P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10c 10b 5a 10a 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 A -A A -A -1 1 1 -1 X.6 1 -1 1 1 -1 -1 -A A -A A -1 1 1 -1 X.7 1 1 1 -1 -1 -1 A A -A -A 1 1 1 1 X.8 1 1 1 -1 -1 -1 -A -A A A 1 1 1 1 X.9 2 . -2 . -2 2 . . . . . -2 2 . X.10 2 . -2 . 2 -2 . . . . . -2 2 . X.11 4 -4 4 . . . . . . . 1 -1 -1 1 X.12 4 4 4 . . . . . . . -1 -1 -1 -1 X.13 4 . -4 . . . . . . . B 1 -1 -B X.14 4 . -4 . . . . . . . -B 1 -1 B A = -E(4) = -Sqrt(-1) = -i B = -E(5)+E(5)^2+E(5)^3-E(5)^4 = -Sqrt(5) = -r5