Galois group: 41T5

• Label: 41T5
• Degree: $41$
• Order: $328$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{41}:C_{8}$

Group action invariants

Degree $n$:		$41$
Transitive number $t$:		$5$
Group:		$C_{41}:C_{8}$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,27,32,3,40,14,9,38)(2,13,23,6,39,28,18,35)(4,26,5,12,37,15,36,29)(7,25,19,21,34,16,22,20)(8,11,10,24,33,30,31,17), (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)
$2$:	$C_2$
$4$:	$C_4$
$8$:	$C_8$

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	Index	Representative
1A	$1^{41}$	$1$	$1$	$0$	$()$
2A	$2^{20},1$	$41$	$2$	$20$	$( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$
4A1	$4^{10},1$	$41$	$4$	$30$	$( 2,33,41,10)( 3,24,40,19)( 4,15,39,28)( 5, 6,38,37)( 7,29,36,14)( 8,20,35,23)( 9,11,34,32)(12,25,31,18)(13,16,30,27)(17,21,26,22)$
4A-1	$4^{10},1$	$41$	$4$	$30$	$( 2,10,41,33)( 3,19,40,24)( 4,28,39,15)( 5,37,38, 6)( 7,14,36,29)( 8,23,35,20)( 9,32,34,11)(12,18,31,25)(13,27,30,16)(17,22,26,21)$
8A1	$8^{5},1$	$41$	$8$	$35$	$( 2,28,33, 4,41,15,10,39)( 3,14,24, 7,40,29,19,36)( 5,27, 6,13,38,16,37,30)( 8,26,20,22,35,17,23,21)( 9,12,11,25,34,31,32,18)$
8A-1	$8^{5},1$	$41$	$8$	$35$	$( 2,39,10,15,41, 4,33,28)( 3,36,19,29,40, 7,24,14)( 5,30,37,16,38,13, 6,27)( 8,21,23,17,35,22,20,26)( 9,18,32,31,34,25,11,12)$
8A3	$8^{5},1$	$41$	$8$	$35$	$( 2, 4,10,28,41,39,33,15)( 3, 7,19,14,40,36,24,29)( 5,13,37,27,38,30, 6,16)( 8,22,23,26,35,21,20,17)( 9,25,32,12,34,18,11,31)$
8A-3	$8^{5},1$	$41$	$8$	$35$	$( 2,15,33,39,41,28,10, 4)( 3,29,24,36,40,14,19, 7)( 5,16, 6,30,38,27,37,13)( 8,17,20,21,35,26,23,22)( 9,31,11,18,34,12,32,25)$
41A1	$41$	$8$	$41$	$40$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$
41A2	$41$	$8$	$41$	$40$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
41A4	$41$	$8$	$41$	$40$	$( 1,25, 8,32,15,39,22, 5,29,12,36,19, 2,26, 9,33,16,40,23, 6,30,13,37,20, 3,27,10,34,17,41,24, 7,31,14,38,21, 4,28,11,35,18)$
41A7	$41$	$8$	$41$	$40$	$( 1,22, 2,23, 3,24, 4,25, 5,26, 6,27, 7,28, 8,29, 9,30,10,31,11,32,12,33,13,34,14,35,15,36,16,37,17,38,18,39,19,40,20,41,21)$
41A8	$41$	$8$	$41$	$40$	$( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15,19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$

Malle's constant $a(G)$:     $1/20$

Group invariants

Order:		$328=2^{3} \cdot 41$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		328.12
Character table:

		1A	2A	4A1	4A-1	8A1	8A-1	8A3	8A-3	41A1	41A2	41A4	41A7	41A8
	Size	1	41	41	41	41	41	41	41	8	8	8	8	8
	2 P	1A	1A	2A	2A	4A1	4A-1	4A-1	4A1	41A8	41A4	41A1	41A2	41A7
	41 P	1A	2A	4A1	4A-1	8A-3	8A3	8A-1	8A1	41A7	41A8	41A2	41A4	41A1
	Type
328.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
328.12.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
328.12.1c1	C	$1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$
328.12.1c2	C	$1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$
328.12.1d1	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	${\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$	$-{\zeta }_8^3$	$1$	$1$	$1$	$1$	$1$
328.12.1d2	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-{\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$	${\zeta }_8$	$1$	$1$	$1$	$1$	$1$
328.12.1d3	C	$1$	$-1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-{\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$	${\zeta }_8^3$	$1$	$1$	$1$	$1$	$1$
328.12.1d4	C	$1$	$-1$	${\zeta }_8^2$	$-{\zeta }_8^2$	${\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$	$-{\zeta }_8$	$1$	$1$	$1$	$1$	$1$
328.12.8a1	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$
328.12.8a2	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$
328.12.8a3	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$
328.12.8a4	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$
328.12.8a5	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$