Properties

Label 40T50
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times D_8$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $50$
Group :  $C_5\times D_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,23,8,28,9,31,13,34,18,39)(2,24,7,27,10,32,14,33,17,40)(3,21,6,26,11,30,15,36,20,37)(4,22,5,25,12,29,16,35,19,38), (1,21,6,28,10,29,16,33,18,37,3,23,7,25,12,32,13,36,20,39,2,22,5,27,9,30,15,34,17,38,4,24,8,26,11,31,14,35,19,40)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $C_5$

Degree 8: $D_{8}$

Degree 10: $C_{10}$

Degree 20: 20T12

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 25]
Character table: Data not available.