Properties

Label 40T23
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{20}:C_2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$
10:  $D_{5}$
16:  $Q_8:C_2$
20:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8:C_2$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 38]
Character table: Data not available.