Galois group: 41T4

• Label: 41T4
• Degree: $41$
• Order: $205$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{41}:C_{5}$

Group action invariants

Degree $n$:		$41$
Transitive number $t$:		$4$
Group:		$C_{41}:C_{5}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,10,18,16,37)(2,20,36,32,33)(3,30,13,7,29)(4,40,31,23,25)(5,9,8,39,21)(6,19,26,14,17)(11,28,34,12,38)(15,27,24,35,22), (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)
$5$:	$C_5$

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

Group invariants

Order:		$205=5 \cdot 41$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		205.1
Character table:

		1A	5A1	5A-1	5A2	5A-2	41A1	41A-1	41A2	41A-2	41A3	41A-3	41A6	41A-6
	Size	1	41	41	41	41	5	5	5	5	5	5	5	5
	5 P	1A	5A-1	5A1	5A2	5A-2	41A6	41A1	41A3	41A2	41A-6	41A-2	41A-1	41A-3
	41 P	1A	1A	1A	1A	1A	41A-6	41A-1	41A-3	41A-2	41A6	41A2	41A1	41A3
	Type
205.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b1	C	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b2	C	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b3	C	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.1b4	C	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
205.1.5a1	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$
205.1.5a2	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$
205.1.5a3	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$
205.1.5a4	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$
205.1.5a5	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$
205.1.5a6	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$
205.1.5a7	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$
205.1.5a8	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-2}+{\zeta }_{41}^5+{\zeta }_{41}^8+{\zeta }_{41}^9$	${\zeta }_{41}^{-9}+{\zeta }_{41}^{-8}+{\zeta }_{41}^{-5}+{\zeta }_{41}^2+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-4}+{\zeta }_{41}+{\zeta }_{41}^{10}+{\zeta }_{41}^{16}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-1}+{\zeta }_{41}^4$	${\zeta }_{41}^{-19}+{\zeta }_{41}^{-17}+{\zeta }_{41}^{-14}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^6+{\zeta }_{41}^{14}+{\zeta }_{41}^{17}+{\zeta }_{41}^{19}$	${\zeta }_{41}^{-12}+{\zeta }_{41}^{-11}+{\zeta }_{41}^3+{\zeta }_{41}^7+{\zeta }_{41}^{13}$	${\zeta }_{41}^{-13}+{\zeta }_{41}^{-7}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{11}+{\zeta }_{41}^{12}$