Galois group: 31T4

• Label: 31T4
• Degree: $31$
• Order: $155$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{31}:C_{5}$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$5$:	$C_5$

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

Group invariants

Order:		$155=5 \cdot 31$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		155.1
Character table:

		1A	5A1	5A-1	5A2	5A-2	31A1	31A-1	31A3	31A-3	31A5	31A-5
	Size	1	31	31	31	31	5	5	5	5	5	5
	5 P	1A	5A-1	5A1	5A2	5A-2	31A1	31A5	31A3	31A-1	31A-3	31A-5
	31 P	1A	5A1	5A-1	5A-2	5A2	31A3	31A-1	31A5	31A-3	31A-5	31A1
	Type
155.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
155.1.1b1	C	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$
155.1.1b2	C	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$	$1$	$1$	$1$	$1$	$1$	$1$
155.1.1b3	C	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$	$1$	$1$	$1$	$1$	$1$	$1$
155.1.1b4	C	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$	$1$	$1$	$1$	$1$	$1$	$1$
155.1.5a1	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$
155.1.5a2	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$
155.1.5a3	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$
155.1.5a4	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$
155.1.5a5	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$
155.1.5a6	C	$5$	$0$	$0$	$0$	$0$	${\zeta }_{31}^{-15}+{\zeta }_{31}+{\zeta }_{31}^2+{\zeta }_{31}^4+{\zeta }_{31}^8$	${\zeta }_{31}^{-8}+{\zeta }_{31}^{-4}+{\zeta }_{31}^{-2}+{\zeta }_{31}^{-1}+{\zeta }_{31}^{15}$	${\zeta }_{31}^{-14}+{\zeta }_{31}^{-7}+{\zeta }_{31}^3+{\zeta }_{31}^6+{\zeta }_{31}^{12}$	${\zeta }_{31}^{-12}+{\zeta }_{31}^{-6}+{\zeta }_{31}^{-3}+{\zeta }_{31}^7+{\zeta }_{31}^{14}$	${\zeta }_{31}^{-13}+{\zeta }_{31}^{-11}+{\zeta }_{31}^5+{\zeta }_{31}^9+{\zeta }_{31}^{10}$	${\zeta }_{31}^{-10}+{\zeta }_{31}^{-9}+{\zeta }_{31}^{-5}+{\zeta }_{31}^{11}+{\zeta }_{31}^{13}$