Galois group: 31T5

• Label: 31T5
• Degree: $31$
• Order: $186$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{31}:C_{6}$

Group action invariants

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

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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $	$31$	$3$	$( 2, 6,26)( 3,11,20)( 4,16,14)( 5,21, 8)( 7,31,27)( 9,10,15)(12,25,28) (13,30,22)(17,19,29)(18,24,23)$
	$ 6, 6, 6, 6, 6, 1 $	$31$	$6$	$( 2, 7, 6,31,26,27)( 3,13,11,30,20,22)( 4,19,16,29,14,17)( 5,25,21,28, 8,12) ( 9,18,10,24,15,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $	$31$	$3$	$( 2,26, 6)( 3,20,11)( 4,14,16)( 5, 8,21)( 7,27,31)( 9,15,10)(12,28,25) (13,22,30)(17,29,19)(18,23,24)$
	$ 6, 6, 6, 6, 6, 1 $	$31$	$6$	$( 2,27,26,31, 6, 7)( 3,22,20,30,11,13)( 4,17,14,29,16,19)( 5,12, 8,28,21,25) ( 9,23,15,24,10,18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$31$	$2$	$( 2,31)( 3,30)( 4,29)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$
	$ 31 $	$6$	$31$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$
	$ 31 $	$6$	$31$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31, 2, 4, 6, 8,10,12,14,16,18, 20,22,24,26,28,30)$
	$ 31 $	$6$	$31$	$( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11, 14,17,20,23,26,29)$
	$ 31 $	$6$	$31$	$( 1, 5, 9,13,17,21,25,29, 2, 6,10,14,18,22,26,30, 3, 7,11,15,19,23,27,31, 4, 8,12,16,20,24,28)$
	$ 31 $	$6$	$31$	$( 1, 9,17,25, 2,10,18,26, 3,11,19,27, 4,12,20,28, 5,13,21,29, 6,14,22,30, 7, 15,23,31, 8,16,24)$

Group invariants

Order:		$186=2 \cdot 3 \cdot 31$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		186.1
Character table:

		1A	2A	3A1	3A-1	6A1	6A-1	31A1	31A2	31A3	31A4	31A8
	Size	1	31	31	31	31	31	6	6	6	6	6
	2 P	1A	1A	3A-1	3A1	3A1	3A-1	31A3	31A1	31A8	31A2	31A4
	3 P	1A	2A	1A	1A	2A	2A	31A4	31A8	31A2	31A3	31A1
	31 P	1A	2A	3A-1	3A1	6A-1	6A1	31A8	31A3	31A4	31A1	31A2
	Type
186.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
186.1.1b	R	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$
186.1.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$
186.1.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$
186.1.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$
186.1.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$1$	$1$	$1$	$1$	$1$
186.1.6a1	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}+{\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-11}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^4+{\zeta }_{31}^7+{\zeta }_{31}^{11}$	${\zeta }_{31}^{-6}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^5+{\zeta }_{31}^6$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^2+{\zeta }_{31}^{10}+{\zeta }_{31}^{12}$
186.1.6a2	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-6}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^5+{\zeta }_{31}^6$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}+{\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^2+{\zeta }_{31}^{10}+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-11}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^4+{\zeta }_{31}^7+{\zeta }_{31}^{11}$
186.1.6a3	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^2+{\zeta }_{31}^{10}+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-11}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^4+{\zeta }_{31}^7+{\zeta }_{31}^{11}$	${\zeta }_{31}^{-6}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^5+{\zeta }_{31}^6$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}+{\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$
186.1.6a4	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-11}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^4+{\zeta }_{31}^7+{\zeta }_{31}^{11}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}+{\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^2+{\zeta }_{31}^{10}+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-6}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^5+{\zeta }_{31}^6$
186.1.6a5	R	$6$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-6}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^5+{\zeta }_{31}^6$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-2}+{\zeta }_{31}^2+{\zeta }_{31}^{10}+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-13}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^{13}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-11}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-4}+{\zeta }_{31}^4+{\zeta }_{31}^7+{\zeta }_{31}^{11}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-8}+{\zeta }_{31}^8+{\zeta }_{31}^9+{\zeta }_{31}^{14}$