Galois group: 41T3

• Label: 41T3
• Degree: $41$
• Order: $164$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{41}:C_{4}$

Group action invariants

Degree $n$:		$41$
Transitive number $t$:		$3$
Group:		$C_{41}:C_{4}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,32,40,9)(2,23,39,18)(3,14,38,27)(4,5,37,36)(6,28,35,13)(7,19,34,22)(8,10,33,31)(11,24,30,17)(12,15,29,26)(16,20,25,21), (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)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

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

Label	Cycle Type	Size	Order	Representative
	$ 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 $	$1$	$1$	$()$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $	$41$	$4$	$( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20) ( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1 $	$41$	$4$	$( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23) ( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$41$	$2$	$( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$
	$ 41 $	$4$	$41$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
	$ 41 $	$4$	$41$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$
	$ 41 $	$4$	$41$	$( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$
	$ 41 $	$4$	$41$	$( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15, 19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$
	$ 41 $	$4$	$41$	$( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$
	$ 41 $	$4$	$41$	$( 1, 8,15,22,29,36, 2, 9,16,23,30,37, 3,10,17,24,31,38, 4,11,18,25,32,39, 5, 12,19,26,33,40, 6,13,20,27,34,41, 7,14,21,28,35)$
	$ 41 $	$4$	$41$	$( 1, 9,17,25,33,41, 8,16,24,32,40, 7,15,23,31,39, 6,14,22,30,38, 5,13,21,29, 37, 4,12,20,28,36, 3,11,19,27,35, 2,10,18,26,34)$
	$ 41 $	$4$	$41$	$( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19, 30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$
	$ 41 $	$4$	$41$	$( 1,13,25,37, 8,20,32, 3,15,27,39,10,22,34, 5,17,29,41,12,24,36, 7,19,31, 2, 14,26,38, 9,21,33, 4,16,28,40,11,23,35, 6,18,30)$
	$ 41 $	$4$	$41$	$( 1,17,33, 8,24,40,15,31, 6,22,38,13,29, 4,20,36,11,27, 2,18,34, 9,25,41,16, 32, 7,23,39,14,30, 5,21,37,12,28, 3,19,35,10,26)$

Group invariants

Order:		$164=2^{2} \cdot 41$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		164.3
Character table:

1A

2A

4A1

4A-1

41A1

41A2

41A3

41A4

41A6

41A7

41A8

41A11

41A12

41A16

Size

1

41

41

41

4

4

4

4

4

4

4

4

4

4

2 P

1A

1A

2A

2A

41A11

41A8

41A6

41A1

41A12

41A3

41A7

41A2

41A4

41A16

41 P

1A

2A

4A1

4A-1

41A7

41A16

41A12

41A2

41A11

41A6

41A3

41A4

41A8

41A1

Type

164.3.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

164.3.1b

R

$1$

$1$

$-1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

164.3.1c1

C

$1$

$-1$

$-i$

$i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

164.3.1c2

C

$1$

$-1$

$i$

$-i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

164.3.4a1

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

164.3.4a2

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

164.3.4a3

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

164.3.4a4

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

164.3.4a5

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

164.3.4a6

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

164.3.4a7

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

164.3.4a8

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

164.3.4a9

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

164.3.4a10

R

$4$

$0$

$0$

$0$

${\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5$

${\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}$

${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$

${\zeta }_{41}^{-20}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{16}+{\zeta }_{41}^{20}$

${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$

${\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^6+{\zeta }_{41}^{13}$

${\zeta }_{41}^{-9}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^9$

${\zeta }_{41}^{-14}+{\zeta }_{41}^{-3}+{\zeta }_{41}^3+{\zeta }_{41}^{14}$

${\zeta }_{41}^{-19}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{19}$

${\zeta }_{41}^{-18}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^{18}$