Galois group: 29T3

• Label: 29T3
• Degree: $29$
• Order: $116$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{29}:C_{4}$

Group action invariants

Degree $n$:		$29$
Transitive number $t$:		$3$
Group:		$C_{29}:C_{4}$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(8,15,22,29), (1,2,3,7,4,24,8,14,5,12,25,27,9,20,15,29,6,23,13,11,26,19,28,22,10,18,21,17,16)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

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 $	$1$	$1$	$()$
	$ 4, 4, 4, 4, 4, 4, 4, 1 $	$29$	$4$	$( 2, 9,16,23)( 3,10,17,24)( 4,11,18,25)( 5,12,19,26)( 6,13,20,27)( 7,14,21,28) ( 8,15,22,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$29$	$2$	$( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,23)(10,24)(11,25)(12,26) (13,27)(14,28)(15,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 1 $	$29$	$4$	$( 2,23,16, 9)( 3,24,17,10)( 4,25,18,11)( 5,26,19,12)( 6,27,20,13)( 7,28,21,14) ( 8,29,22,15)$
	$ 29 $	$4$	$29$	$( 1, 2, 3, 7, 4,24, 8,14, 5,12,25,27, 9,20,15,29, 6,23,13,11,26,19,28,22,10, 18,21,17,16)$
	$ 29 $	$4$	$29$	$( 1, 3, 4, 8, 5,25, 9,15, 6,13,26,28,10,21,16, 2, 7,24,14,12,27,20,29,23,11, 19,22,18,17)$
	$ 29 $	$4$	$29$	$( 1, 4, 5, 9, 6,26,10,16, 7,14,27,29,11,22,17, 3, 8,25,15,13,28,21, 2,24,12, 20,23,19,18)$
	$ 29 $	$4$	$29$	$( 1, 5, 6,10, 7,27,11,17, 8,15,28, 2,12,23,18, 4, 9,26,16,14,29,22, 3,25,13, 21,24,20,19)$
	$ 29 $	$4$	$29$	$( 1, 6, 7,11, 8,28,12,18, 9,16,29, 3,13,24,19, 5,10,27,17,15, 2,23, 4,26,14, 22,25,21,20)$
	$ 29 $	$4$	$29$	$( 1, 7, 8,12, 9,29,13,19,10,17, 2, 4,14,25,20, 6,11,28,18,16, 3,24, 5,27,15, 23,26,22,21)$
	$ 29 $	$4$	$29$	$( 1, 8, 9,13,10, 2,14,20,11,18, 3, 5,15,26,21, 7,12,29,19,17, 4,25, 6,28,16, 24,27,23,22)$

Group invariants

Order:		$116=2^{2} \cdot 29$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		116.3
Character table:

		1A	2A	4A1	4A-1	29A1	29A2	29A3	29A4	29A6	29A8	29A11
	Size	1	29	29	29	4	4	4	4	4	4	4
	2 P	1A	1A	2A	2A	29A1	29A2	29A11	29A6	29A3	29A4	29A8
	29 P	1A	2A	4A-1	4A1	29A11	29A3	29A2	29A8	29A4	29A6	29A1
	Type
116.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
116.3.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
116.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
116.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
116.3.4a1	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$
116.3.4a2	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$
116.3.4a3	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$
116.3.4a4	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$
116.3.4a5	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$
116.3.4a6	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$
116.3.4a7	R	$4$	$0$	$0$	$0$	${\zeta }_{29}^{-5}+{\zeta }_{29}^{-2}+{\zeta }_{29}^2+{\zeta }_{29}^5$	${\zeta }_{29}^{-10}+{\zeta }_{29}^{-4}+{\zeta }_{29}^4+{\zeta }_{29}^{10}$	${\zeta }_{29}^{-14}+{\zeta }_{29}^{-6}+{\zeta }_{29}^6+{\zeta }_{29}^{14}$	${\zeta }_{29}^{-9}+{\zeta }_{29}^{-8}+{\zeta }_{29}^8+{\zeta }_{29}^9$	${\zeta }_{29}^{-12}+{\zeta }_{29}^{-1}+{\zeta }_{29}+{\zeta }_{29}^{12}$	${\zeta }_{29}^{-13}+{\zeta }_{29}^{-11}+{\zeta }_{29}^{11}+{\zeta }_{29}^{13}$	${\zeta }_{29}^{-7}+{\zeta }_{29}^{-3}+{\zeta }_{29}^3+{\zeta }_{29}^7$