Galois group: 41T4

• Label: 41T4
• Degree: $41$
• Order: $205$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Transitivity: $1$
• Primitive: yes
• $p$-group: no
• Group: $C_{41}:C_{5}$

Group invariants

Abstract group:		$C_{41}:C_{5}$
Order:		$205=5 \cdot 41$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$41$
Transitive number $t$:		$4$
Parity:		$1$
Transitivity:		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

$\card{(G/N)}$	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	Index	Representative
1A	$1^{41}$	$1$	$1$	$0$	$()$
5A1	$5^{8},1$	$41$	$5$	$32$	$( 1,24, 8,12,11)( 2,34,26,28, 7)( 4,13,21,19,40)( 5,23,39,35,36)( 6,33,16,10,32)( 9,22,29,17,20)(14,31,37,15,41)(18,30,27,38,25)$
5A-1	$5^{8},1$	$41$	$5$	$32$	$( 1,11,12, 8,24)( 2, 7,28,26,34)( 4,40,19,21,13)( 5,36,35,39,23)( 6,32,10,16,33)( 9,20,17,29,22)(14,41,15,37,31)(18,25,38,27,30)$
5A2	$5^{8},1$	$41$	$5$	$32$	$( 1, 8,11,24,12)( 2,26, 7,34,28)( 4,21,40,13,19)( 5,39,36,23,35)( 6,16,32,33,10)( 9,29,20,22,17)(14,37,41,31,15)(18,27,25,30,38)$
5A-2	$5^{8},1$	$41$	$5$	$32$	$( 1,12,24,11, 8)( 2,28,34, 7,26)( 4,19,13,40,21)( 5,35,23,36,39)( 6,10,33,32,16)( 9,17,22,20,29)(14,15,31,41,37)(18,38,30,25,27)$
41A1	$41$	$5$	$41$	$40$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
41A-1	$41$	$5$	$41$	$40$	$( 1, 5, 9,13,17,21,25,29,33,37,41, 4, 8,12,16,20,24,28,32,36,40, 3, 7,11,15,19,23,27,31,35,39, 2, 6,10,14,18,22,26,30,34,38)$
41A2	$41$	$5$	$41$	$40$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$
41A-2	$41$	$5$	$41$	$40$	$( 1, 6,11,16,21,26,31,36,41, 5,10,15,20,25,30,35,40, 4, 9,14,19,24,29,34,39, 3, 8,13,18,23,28,33,38, 2, 7,12,17,22,27,32,37)$
41A3	$41$	$5$	$41$	$40$	$( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$
41A-3	$41$	$5$	$41$	$40$	$( 1,12,23,34, 4,15,26,37, 7,18,29,40,10,21,32, 2,13,24,35, 5,16,27,38, 8,19,30,41,11,22,33, 3,14,25,36, 6,17,28,39, 9,20,31)$
41A6	$41$	$5$	$41$	$40$	$( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22,28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$
41A-6	$41$	$5$	$41$	$40$	$( 1,16,31, 5,20,35, 9,24,39,13,28, 2,17,32, 6,21,36,10,25,40,14,29, 3,18,33, 7,22,37,11,26,41,15,30, 4,19,34, 8,23,38,12,27)$

Malle's constant $a(G)$:     $1/32$

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

Regular extensions

Data not computed