Galois group: 20T4

• Label: 20T4
• Degree: $20$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{10}$

Group invariants

Abstract group:		$D_{10}$
Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$10$:	$D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 10: $D_5$, $D_{10}$ x 2

Low degree siblings

10T3 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{20}$	$1$	$1$	$0$	$()$
2A	$2^{10}$	$1$	$2$	$10$	$( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,20)(10,19)$
2B	$2^{10}$	$5$	$2$	$10$	$( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(17,19)(18,20)$
2C	$2^{10}$	$5$	$2$	$10$	$( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$
5A1	$5^{4}$	$2$	$5$	$16$	$( 1,10,18, 6,14)( 2, 9,17, 5,13)( 3,11,20, 7,15)( 4,12,19, 8,16)$
5A2	$5^{4}$	$2$	$5$	$16$	$( 1,18,14,10, 6)( 2,17,13, 9, 5)( 3,20,15,11, 7)( 4,19,16,12, 8)$
10A1	$10^{2}$	$2$	$10$	$18$	$( 1,16,10, 4,18,12, 6,19,14, 8)( 2,15, 9, 3,17,11, 5,20,13, 7)$
10A3	$10^{2}$	$2$	$10$	$18$	$( 1, 4, 6, 8,10,12,14,16,18,19)( 2, 3, 5, 7, 9,11,13,15,17,20)$

Malle's constant $a(G)$:     $1/10$

Character table

		1A	2A	2B	2C	5A1	5A2	10A1	10A3
	Size	1	1	5	5	2	2	2	2
	2 P	1A	1A	1A	1A	5A2	5A1	5A1	5A2
	5 P	1A	2A	2B	2C	1A	1A	2A	2A
	Type
20.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
20.4.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
20.4.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
20.4.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
20.4.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
20.4.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
20.4.2b1	R	$2$	$-2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
20.4.2b2	R	$2$	$-2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$

Regular extensions

Data not computed