Properties

Label 41T7
Order \(820\)
n \(41\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{41}:C_{20}$

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Group action invariants

Degree $n$ :  $41$
Transitive number $t$ :  $7$
Group :  $C_{41}:C_{20}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,36,25,39,10,32,4,21,18,33,40,5,16,2,31,9,37,20,23,8)(3,26,34,35,30,14,12,22,13,17,38,15,7,6,11,27,29,19,28,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$
20:  20T1

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
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 $ $1$ $1$ $()$
$ 41 $ $20$ $41$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
$ 41 $ $20$ $41$ $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14, 39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31, 39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $41$ $2$ $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8, 39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30, 39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$
$ 20, 20, 1 $ $41$ $20$ $( 2,37,26,40,11,33, 5,22,19,34,41, 6,17, 3,32,10,38,21,24, 9)( 4,27,35,36,31, 15,13,23,14,18,39,16, 8, 7,12,28,30,20,29,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ $41$ $4$ $( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23) ( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$
$ 20, 20, 1 $ $41$ $20$ $( 2,34,24,22,38,33,32,40,17,37,41, 9,19,21, 5,10,11, 3,26, 6)( 4,18,29,23,30, 15,12,36, 8,27,39,25,14,20,13,28,31, 7,35,16)$
$ 20, 20, 1 $ $41$ $20$ $( 2,21,32, 6,19,33,26, 9,38, 3,41,22,11,37,24,10,17,34, 5,40)( 4,20,12,16,14, 15,35,25,30, 7,39,23,31,27,29,28, 8,18,13,36)$
$ 20, 20, 1 $ $41$ $20$ $( 2, 3, 5, 9,17,33,24, 6,11,21,41,40,38,34,26,10,19,37,32,22)( 4, 7,13,25, 8, 15,29,16,31,20,39,36,30,18,35,28,14,27,12,23)$
$ 20, 20, 1 $ $41$ $20$ $( 2,40, 5,34,17,10,24,37,11,22,41, 3,38, 9,26,33,19, 6,32,21)( 4,36,13,18, 8, 28,29,27,31,23,39, 7,30,25,35,15,14,16,12,20)$
$ 20, 20, 1 $ $41$ $20$ $( 2,22,32,37,19,10,26,34,38,40,41,21,11, 6,24,33,17, 9, 5, 3)( 4,23,12,27,14, 28,35,18,30,36,39,20,31,16,29,15, 8,25,13, 7)$
$ 20, 20, 1 $ $41$ $20$ $( 2, 6,26, 3,11,10, 5,21,19, 9,41,37,17,40,32,33,38,22,24,34)( 4,16,35, 7,31, 28,13,20,14,25,39,27, 8,36,12,15,30,23,29,18)$
$ 20, 20, 1 $ $41$ $20$ $( 2, 9,24,21,38,10,32, 3,17, 6,41,34,19,22, 5,33,11,40,26,37)( 4,25,29,20,30, 28,12, 7, 8,16,39,18,14,23,13,15,31,36,35,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $ $41$ $4$ $( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20) ( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$

Group invariants

Order:  $820=2^{2} \cdot 5 \cdot 41$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [820, 7]
Character table: Data not available.