# Properties

 Label 40T13 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{10}.C_4$

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $13$ Group: $C_{10}.C_4$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $40$ Generators: (1,8,33,29,2,7,34,30)(3,6,36,31,4,5,35,32)(9,24,28,13,10,23,27,14)(11,22,25,16,12,21,26,15)(17,38,20,39,18,37,19,40), (1,27,11,36,20)(2,28,12,35,19)(3,26,10,34,17)(4,25,9,33,18)(5,29,16,37,24)(6,30,15,38,23)(7,32,13,40,21)(8,31,14,39,22)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$20$:  $F_5$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 8: $C_8$

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [40, 3]
 Character table:  2 3 3 3 3 3 3 3 3 1 1 5 1 1 . . . . . . 1 1 1a 2a 4a 4b 8a 8b 8c 8d 5a 10a 2P 1a 1a 2a 2a 4a 4a 4b 4b 5a 5a 3P 1a 2a 4b 4a 8d 8c 8b 8a 5a 10a 5P 1a 2a 4a 4b 8b 8a 8d 8c 1a 2a 7P 1a 2a 4b 4a 8c 8d 8a 8b 5a 10a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 1 X.3 1 -1 A -A B -B /B -/B 1 -1 X.4 1 -1 A -A -B B -/B /B 1 -1 X.5 1 -1 -A A -/B /B -B B 1 -1 X.6 1 -1 -A A /B -/B B -B 1 -1 X.7 1 1 -1 -1 A A -A -A 1 1 X.8 1 1 -1 -1 -A -A A A 1 1 X.9 4 -4 . . . . . . -1 1 X.10 4 4 . . . . . . -1 -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8)^3