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 invariants

Abstract group:		$C_{41}:C_{8}$
Order:		$328=2^{3} \cdot 41$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$41$
Transitive number $t$:		$5$
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

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

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	41A2	41A4	41A8	41A1	41A7
	41 P	1A	2A	4A1	4A-1	8A-3	8A3	8A-1	8A1	41A4	41A8	41A7	41A2	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	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$
328.12.8a2	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$
328.12.8a3	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$
328.12.8a4	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$
328.12.8a5	R	$8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{41}^{-14}+{\zeta }_{41}^{-9}+{\zeta }_{41}^{-3}+{\zeta }_{41}^{-1}+{\zeta }_{41}+{\zeta }_{41}^3+{\zeta }_{41}^9+{\zeta }_{41}^{14}$	${\zeta }_{41}^{-20}+{\zeta }_{41}^{-19}+{\zeta }_{41}^{-16}+{\zeta }_{41}^{-7}+{\zeta }_{41}^7+{\zeta }_{41}^{16}+{\zeta }_{41}^{19}+{\zeta }_{41}^{20}$	${\zeta }_{41}^{-15}+{\zeta }_{41}^{-12}+{\zeta }_{41}^{-5}+{\zeta }_{41}^{-4}+{\zeta }_{41}^4+{\zeta }_{41}^5+{\zeta }_{41}^{12}+{\zeta }_{41}^{15}$	${\zeta }_{41}^{-17}+{\zeta }_{41}^{-11}+{\zeta }_{41}^{-10}+{\zeta }_{41}^{-8}+{\zeta }_{41}^8+{\zeta }_{41}^{10}+{\zeta }_{41}^{11}+{\zeta }_{41}^{17}$	${\zeta }_{41}^{-18}+{\zeta }_{41}^{-13}+{\zeta }_{41}^{-6}+{\zeta }_{41}^{-2}+{\zeta }_{41}^2+{\zeta }_{41}^6+{\zeta }_{41}^{13}+{\zeta }_{41}^{18}$

Regular extensions

Data not computed