Properties

Label 40T37
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 $t$ :  $37$
Group :  $D_{20}:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,2,7)(3,5,4,6)(9,38,10,37)(11,40,12,39)(13,34,14,33)(15,36,16,35)(17,30,18,29)(19,32,20,31)(21,27,22,28)(23,25,24,26), (1,29)(2,30)(3,32)(4,31)(5,25)(6,26)(7,28)(8,27)(9,23)(10,24)(11,22)(12,21)(13,19)(14,20)(15,17)(16,18)(33,39)(34,40)(35,38)(36,37), (1,15,25,39,12,24,35,8,20,31)(2,16,26,40,11,23,36,7,19,32)(3,14,28,38,9,21,34,5,18,29)(4,13,27,37,10,22,33,6,17,30)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Degree 20: 20T8

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

Group invariants

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