Properties

Label 40T38
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_{20}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
10:  $D_{5}$
16:  $D_4\times C_2$
20:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $D_4\times C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8, $D_{20}$ 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 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, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 5,38)( 6,37)( 7,40)( 8,39)( 9,34)(10,33)(11,35)(12,36)(13,30)(14,29) (15,32)(16,31)(17,27)(18,28)(19,25)(20,26)(23,24)$
$ 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 $ $10$ $2$ $( 1, 3)( 2, 4)( 5,40)( 6,39)( 7,37)( 8,38)( 9,36)(10,35)(11,34)(12,33)(13,31) (14,32)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,24)(22,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,39,38,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,39)(10,40)(11,37)(12,38)(13,36)(14,35)(15,34) (16,33)(17,32)(18,31)(19,29)(20,30)(21,25)(22,26)(23,27)(24,28)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 5,12,13,20,22,26,30,36,38)( 2, 6,11,14,19,21,25,29,35,37)( 3, 8, 9,15,17, 23,28,31,33,40)( 4, 7,10,16,18,24,27,32,34,39)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 6,12,14,20,21,26,29,36,37)( 2, 5,11,13,19,22,25,30,35,38)( 3, 7, 9,16,17, 24,28,32,33,39)( 4, 8,10,15,18,23,27,31,34,40)$
$ 20, 20 $ $2$ $20$ $( 1, 7,11,15,20,24,25,31,36,39, 2, 8,12,16,19,23,26,32,35,40)( 3, 5,10,14,17, 22,27,29,33,38, 4, 6, 9,13,18,21,28,30,34,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,37)(10,38)(11,40)(12,39)(13,34)(14,33)(15,35) (16,36)(17,29)(18,30)(19,31)(20,32)(21,28)(22,27)(23,25)(24,26)$
$ 20, 20 $ $2$ $20$ $( 1, 8,11,16,20,23,25,32,36,40, 2, 7,12,15,19,24,26,31,35,39)( 3, 6,10,13,17, 21,27,30,33,37, 4, 5, 9,14,18,22,28,29,34,38)$
$ 20, 20 $ $2$ $20$ $( 1, 9,19,27,36, 3,11,18,26,33, 2,10,20,28,35, 4,12,17,25,34)( 5,15,21,32,38, 8,14,24,30,40, 6,16,22,31,37, 7,13,23,29,39)$
$ 20, 20 $ $2$ $20$ $( 1,10,19,28,36, 4,11,17,26,34, 2, 9,20,27,35, 3,12,18,25,33)( 5,16,21,31,38, 7,14,23,30,39, 6,15,22,32,37, 8,13,24,29,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,11,20,25,36, 2,12,19,26,35)( 3,10,17,27,33, 4, 9,18,28,34)( 5,14,22,29,38, 6,13,21,30,37)( 7,15,24,31,39, 8,16,23,32,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,20,26,36)( 2,11,19,25,35)( 3, 9,17,28,33)( 4,10,18,27,34) ( 5,13,22,30,38)( 6,14,21,29,37)( 7,16,24,32,39)( 8,15,23,31,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,13,26,38,12,22,36, 5,20,30)( 2,14,25,37,11,21,35, 6,19,29)( 3,15,28,40, 9, 23,33, 8,17,31)( 4,16,27,39,10,24,34, 7,18,32)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,14,26,37,12,21,36, 6,20,29)( 2,13,25,38,11,22,35, 5,19,30)( 3,16,28,39, 9, 24,33, 7,17,32)( 4,15,27,40,10,23,34, 8,18,31)$
$ 20, 20 $ $2$ $20$ $( 1,15,25,39,12,23,35, 7,20,31, 2,16,26,40,11,24,36, 8,19,32)( 3,14,27,38, 9, 21,34, 5,17,29, 4,13,28,37,10,22,33, 6,18,30)$
$ 20, 20 $ $2$ $20$ $( 1,16,25,40,12,24,35, 8,20,32, 2,15,26,39,11,23,36, 7,19,31)( 3,13,27,37, 9, 22,34, 6,17,30, 4,14,28,38,10,21,33, 5,18,29)$
$ 20, 20 $ $2$ $20$ $( 1,17,35,10,26, 3,19,34,12,28, 2,18,36, 9,25, 4,20,33,11,27)( 5,23,37,16,30, 8,21,39,13,31, 6,24,38,15,29, 7,22,40,14,32)$
$ 20, 20 $ $2$ $20$ $( 1,18,35, 9,26, 4,19,33,12,27, 2,17,36,10,25, 3,20,34,11,28)( 5,24,37,15,30, 7,21,40,13,32, 6,23,38,16,29, 8,22,39,14,31)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,36,11,26, 2,20,35,12,25)( 3,18,33,10,28, 4,17,34, 9,27)( 5,21,38,14,30, 6,22,37,13,29)( 7,23,39,15,32, 8,24,40,16,31)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,20,36,12,26)( 2,19,35,11,25)( 3,17,33, 9,28)( 4,18,34,10,27) ( 5,22,38,13,30)( 6,21,37,14,29)( 7,24,39,16,32)( 8,23,40,15,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,25)( 6,26)( 7,28)( 8,27)( 9,32)(10,31)(11,30) (12,29)(13,35)(14,36)(15,34)(16,33)(17,39)(18,40)(19,38)(20,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,26)( 6,25)( 7,27)( 8,28)( 9,31)(10,32)(11,29) (12,30)(13,36)(14,35)(15,33)(16,34)(17,40)(18,39)(19,37)(20,38)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23, 2,24)( 3,21, 4,22)( 5,28, 6,27)( 7,26, 8,25)( 9,29,10,30)(11,32,12,31) (13,33,14,34)(15,35,16,36)(17,37,18,38)(19,39,20,40)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 37]
Character table: Data not available.