Galois group: 43T4

• Label: 43T4
• Degree: $43$
• Order: $258$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{43}:C_{6}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

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

Group invariants

Order:		$258=2 \cdot 3 \cdot 43$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		258.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	43A1	43A2	43A3	43A4	43A5	43A9	43A10
	Size	1	43	43	43	43	43	6	6	6	6	6	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	43A4	43A10	43A3	43A5	43A1	43A2	43A9
	3 P	1A	2A	1A	1A	2A	2A	43A1	43A4	43A10	43A2	43A9	43A3	43A5
	43 P	1A	2A	3A1	3A-1	6A1	6A-1	43A2	43A5	43A9	43A4	43A3	43A1	43A10
	Type
258.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
258.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$
258.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$
258.1.6a3	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$
258.1.6a4	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$
258.1.6a5	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$
258.1.6a6	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$
258.1.6a7	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{43}^{-7}+{\zeta }_{43}^{-6}+{\zeta }_{43}^{-1}+{\zeta }_{43}+{\zeta }_{43}^6+{\zeta }_{43}^7$	${\zeta }_{43}^{-14}+{\zeta }_{43}^{-12}+{\zeta }_{43}^{-2}+{\zeta }_{43}^2+{\zeta }_{43}^{12}+{\zeta }_{43}^{14}$	${\zeta }_{43}^{-21}+{\zeta }_{43}^{-18}+{\zeta }_{43}^{-3}+{\zeta }_{43}^3+{\zeta }_{43}^{18}+{\zeta }_{43}^{21}$	${\zeta }_{43}^{-19}+{\zeta }_{43}^{-15}+{\zeta }_{43}^{-4}+{\zeta }_{43}^4+{\zeta }_{43}^{15}+{\zeta }_{43}^{19}$	${\zeta }_{43}^{-13}+{\zeta }_{43}^{-8}+{\zeta }_{43}^{-5}+{\zeta }_{43}^5+{\zeta }_{43}^8+{\zeta }_{43}^{13}$	${\zeta }_{43}^{-20}+{\zeta }_{43}^{-11}+{\zeta }_{43}^{-9}+{\zeta }_{43}^9+{\zeta }_{43}^{11}+{\zeta }_{43}^{20}$	${\zeta }_{43}^{-17}+{\zeta }_{43}^{-16}+{\zeta }_{43}^{-10}+{\zeta }_{43}^{10}+{\zeta }_{43}^{16}+{\zeta }_{43}^{17}$