Galois group: 20T2

• Label: 20T2
• Degree: $20$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_5:C_4$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$2$
Group:		$C_5:C_4$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$20$
Generators:		(1,19,2,20)(3,17,4,18)(5,15,6,16)(7,14,8,13)(9,12,10,11), (1,12,2,11)(3,10,4,9)(5,7,6,8)(13,19,14,20)(15,17,16,18)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$10$:	$D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 10: $D_5$

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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 3, 2, 4)( 5,19, 6,20)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 4, 2, 3)( 5,20, 6,19)( 7,18, 8,17)( 9,16,10,15)(11,14,12,13)$
	$ 5, 5, 5, 5 $	$2$	$5$	$( 1, 5, 9,13,18)( 2, 6,10,14,17)( 3, 7,12,15,19)( 4, 8,11,16,20)$
	$ 10, 10 $	$2$	$10$	$( 1, 6, 9,14,18, 2, 5,10,13,17)( 3, 8,12,16,19, 4, 7,11,15,20)$
	$ 5, 5, 5, 5 $	$2$	$5$	$( 1, 9,18, 5,13)( 2,10,17, 6,14)( 3,12,19, 7,15)( 4,11,20, 8,16)$
	$ 10, 10 $	$2$	$10$	$( 1,10,18, 6,13, 2, 9,17, 5,14)( 3,11,19, 8,15, 4,12,20, 7,16)$

Group invariants

Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		20.1
Character table:

		1A	2A	4A1	4A-1	5A1	5A2	10A1	10A3
	Size	1	1	5	5	2	2	2	2
	2 P	1A	1A	2A	2A	5A2	5A1	5A1	5A2
	5 P	1A	2A	4A1	4A-1	1A	1A	2A	2A
	Type
20.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
20.1.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
20.1.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$-1$	$-1$
20.1.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$-1$	$-1$
20.1.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
20.1.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
20.1.2b1	S	$2$	$-2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
20.1.2b2	S	$2$	$-2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$