# Properties

 Label 40T2 Degree $40$ Order $40$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group no Group: $C_2\times C_{20}$

# Related objects

## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $2$ Group: $C_2\times C_{20}$ Parity: $1$ Primitive: no Nilpotency class: $1$ $|\Aut(F/K)|$: $40$ Generators: (1,29,20,5,35,23,11,37,28,13,3,32,18,8,34,22,9,39,25,16)(2,30,19,6,36,24,12,38,27,14,4,31,17,7,33,21,10,40,26,15), (1,10,18,27,35,2,9,17,28,36)(3,12,20,26,34,4,11,19,25,33)(5,15,22,31,37,6,16,21,32,38)(7,13,24,29,40,8,14,23,30,39)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$5$:  $C_5$
$8$:  $C_4\times C_2$
$10$:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 5: $C_5$

Degree 8: $C_4\times C_2$

Degree 10: $C_{10}$ x 3

Degree 20: 20T1 x 2, 20T3

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

## Group invariants

 Order: $40=2^{3} \cdot 5$ Cyclic: no Abelian: yes Solvable: yes GAP id: [40, 9]
 Character table: not available.