Properties

Label 40T1
Order \(40\)
n \(40\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{40}$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
8:  $C_8$
10:  $C_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 8: $C_8$

Degree 10: $C_{10}$

Degree 20: 20T1

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [40, 2]
Character table: Data not available.