Galois group: 43T5

• Label: 43T5
• Degree: $43$
• Order: $301$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{43}:C_{7}$

Group action invariants

Degree $n$:		$43$
Transitive number $t$:		$5$
Group:		$C_{43}:C_{7}$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43), (1,41,4,35,16,11,21)(2,39,8,27,32,22,42)(3,37,12,19,5,33,20)(6,31,24,38,10,23,40)(7,29,28,30,26,34,18)(9,25,36,14,15,13,17)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$7$:	$C_7$

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 $	$1$	$1$	$()$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2, 5,17,22,42,36,12)( 3, 9,33,43,40,28,23)( 4,13, 6,21,38,20,34) ( 7,25,11,41,32,39,24)( 8,29,27,19,30,31,35)(10,37,16,18,26,15,14)$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2,12,36,42,22,17, 5)( 3,23,28,40,43,33, 9)( 4,34,20,38,21, 6,13) ( 7,24,39,32,41,11,25)( 8,35,31,30,19,27,29)(10,14,15,26,18,16,37)$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2,17,42,12, 5,22,36)( 3,33,40,23, 9,43,28)( 4, 6,38,34,13,21,20) ( 7,11,32,24,25,41,39)( 8,27,30,35,29,19,31)(10,16,26,14,37,18,15)$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2,22,12,17,36, 5,42)( 3,43,23,33,28, 9,40)( 4,21,34, 6,20,13,38) ( 7,41,24,11,39,25,32)( 8,19,35,27,31,29,30)(10,18,14,16,15,37,26)$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2,36,22, 5,12,42,17)( 3,28,43, 9,23,40,33)( 4,20,21,13,34,38, 6) ( 7,39,41,25,24,32,11)( 8,31,19,29,35,30,27)(10,15,18,37,14,26,16)$
	$ 7, 7, 7, 7, 7, 7, 1 $	$43$	$7$	$( 2,42, 5,36,17,12,22)( 3,40, 9,28,33,23,43)( 4,38,13,20, 6,34,21) ( 7,32,25,39,11,24,41)( 8,30,29,31,27,35,19)(10,26,37,15,16,14,18)$
	$ 43 $	$7$	$43$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$
	$ 43 $	$7$	$43$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43, 2, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42)$
	$ 43 $	$7$	$43$	$( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40,43, 3, 6, 9,12,15,18,21,24,27,30, 33,36,39,42, 2, 5, 8,11,14,17,20,23,26,29,32,35,38,41)$
	$ 43 $	$7$	$43$	$( 1, 7,13,19,25,31,37,43, 6,12,18,24,30,36,42, 5,11,17,23,29,35,41, 4,10,16, 22,28,34,40, 3, 9,15,21,27,33,39, 2, 8,14,20,26,32,38)$
	$ 43 $	$7$	$43$	$( 1, 8,15,22,29,36,43, 7,14,21,28,35,42, 6,13,20,27,34,41, 5,12,19,26,33,40, 4,11,18,25,32,39, 3,10,17,24,31,38, 2, 9,16,23,30,37)$
	$ 43 $	$7$	$43$	$( 1,10,19,28,37, 3,12,21,30,39, 5,14,23,32,41, 7,16,25,34,43, 9,18,27,36, 2, 11,20,29,38, 4,13,22,31,40, 6,15,24,33,42, 8,17,26,35)$

Group invariants

Order:		$301=7 \cdot 43$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		301.1
Character table:

		1A	7A1	7A-1	7A2	7A-2	7A3	7A-3	43A1	43A-1	43A3	43A-3	43A7	43A-7
	Size	1	43	43	43	43	43	43	7	7	7	7	7	7
	7 P	1A	7A-2	7A-1	7A-3	7A3	7A2	7A1	43A3	43A-3	43A-7	43A7	43A1	43A-1
	43 P	1A	7A-3	7A2	7A-1	7A1	7A3	7A-2	43A7	43A-7	43A1	43A-1	43A-3	43A3
	Type
301.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b1	C	$1$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^{-2}$	${\zeta }_7^2$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b2	C	$1$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^2$	${\zeta }_7^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b3	C	$1$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^3$	${\zeta }_7^{-3}$	${\zeta }_7$	${\zeta }_7^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b4	C	$1$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^{-3}$	${\zeta }_7^3$	${\zeta }_7^{-1}$	${\zeta }_7$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b5	C	$1$	${\zeta }_7^{-1}$	${\zeta }_7$	${\zeta }_7^{-2}$	${\zeta }_7^2$	${\zeta }_7^{-3}$	${\zeta }_7^3$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.1b6	C	$1$	${\zeta }_7$	${\zeta }_7^{-1}$	${\zeta }_7^2$	${\zeta }_7^{-2}$	${\zeta }_7^3$	${\zeta }_7^{-3}$	$1$	$1$	$1$	$1$	$1$	$1$
301.1.7a1	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$
301.1.7a2	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$
301.1.7a3	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$
301.1.7a4	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$
301.1.7a5	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$
301.1.7a6	C	$7$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-8}+{\zeta }_{43}^{-2}+{\zeta }_{43}+{\zeta }_{43}^4+{\zeta }_{43}^{11}+{\zeta }_{43}^{16}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-4}+{\zeta }_{43}^{-1}+{\zeta }_{43}^2+{\zeta }_{43}^8$	${\zeta }_{43}^{-10}+{\zeta }_{43}^{-6}+{\zeta }_{43}^3+{\zeta }_{43}^5+{\zeta }_{43}^{12}+{\zeta }_{43}^{19}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-19}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-5}+{\zeta }_{43}^{-3}+{\zeta }_{43}^6+{\zeta }_{43}^{10}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-14}+{\zeta }_{43}^{-13}+{\zeta }_{43}^{-9}+{\zeta }_{43}^7+{\zeta }_{43}^{18}$	${\zeta }_{43}^{-18}+{\zeta }_{43}^{-7}+{\zeta }_{43}^9+{\zeta }_{43}^{13}+{\zeta }_{43}^{14}+{\zeta }_{43}^{15}+{\zeta }_{43}^{17}$