Galois group: 31T6

• Label: 31T6
• Degree: $31$
• Order: $310$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{31}:C_{10}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$5$:	$C_5$
$10$:	$C_{10}$

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

Group invariants

Order:		$310=2 \cdot 5 \cdot 31$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		310.1
Character table:

		1A	2A	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3	31A1	31A3	31A5
	Size	1	31	31	31	31	31	31	31	31	31	10	10	10
	2 P	1A	1A	5A2	5A-2	5A-1	5A1	5A1	5A-1	5A-2	5A2	31A1	31A3	31A5
	5 P	1A	2A	5A-2	5A2	5A1	5A-1	10A3	10A-3	10A-1	10A1	31A3	31A5	31A1
	31 P	1A	2A	1A	1A	1A	1A	2A	2A	2A	2A	31A5	31A1	31A3
	Type
310.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
310.1.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
310.1.1c1	C	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$
310.1.1c2	C	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$
310.1.1c3	C	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$
310.1.1c4	C	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$
310.1.1d1	C	$1$	$-1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	$1$	$1$	$1$
310.1.1d2	C	$1$	$-1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	$1$	$1$	$1$
310.1.1d3	C	$1$	$-1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	$1$	$1$	$1$
310.1.1d4	C	$1$	$-1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	$1$	$1$	$1$
310.1.10a1	R	$10$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-12}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^7+{\zeta }_{31}^{12}+{\zeta }_{31}^{14}$
310.1.10a2	R	$10$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-12}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^7+{\zeta }_{31}^{12}+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8+{\zeta }_{31}^{15}$
310.1.10a3	R	$10$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-15}+{\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-12}+{\zeta }_{31}^{-7}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^7+{\zeta }_{31}^{12}+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$