Galois group: 20T19

• Label: 20T19
• Degree: $20$
• Order: $80$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $D_{10}:C_4$

Group invariants

Abstract group:		$D_{10}:C_4$
Order:		$80=2^{4} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$16$:	$C_2^2:C_4$
$20$:	$F_5$
$40$:	$F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T19, 20T22 x 2, 40T26, 40T45, 40T55 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	$2^{10}$	$1$	$2$	$10$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B	$2^{10}$	$2$	$2$	$10$	$( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
2C	$2^{10}$	$5$	$2$	$10$	$( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,20)( 9,17)(10,18)(11,15)(12,16)(13,14)$
2D	$2^{8},1^{4}$	$5$	$2$	$8$	$( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)$
2E	$2^{10}$	$10$	$2$	$10$	$( 1,16)( 2,15)( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)(17,20)(18,19)$
4A1	$4^{5}$	$10$	$4$	$15$	$( 1, 7, 6,19)( 2, 8, 5,20)( 3,13, 4,14)( 9,12,17,16)(10,11,18,15)$
4A-1	$4^{5}$	$10$	$4$	$15$	$( 1,19, 6, 7)( 2,20, 5, 8)( 3,14, 4,13)( 9,16,17,12)(10,15,18,11)$
4B1	$4^{4},2,1^{2}$	$10$	$4$	$13$	$( 3,15,20, 8)( 4,16,19, 7)( 5,10,17,14)( 6, 9,18,13)(11,12)$
4B-1	$4^{4},2,1^{2}$	$10$	$4$	$13$	$( 3, 8,20,15)( 4, 7,19,16)( 5,14,17,10)( 6,13,18, 9)(11,12)$
5A	$5^{4}$	$4$	$5$	$16$	$( 1,10,17, 5,14)( 2, 9,18, 6,13)( 3,11,20, 7,16)( 4,12,19, 8,15)$
10A	$10^{2}$	$4$	$10$	$18$	$( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$
10B1	$10^{2}$	$4$	$10$	$18$	$( 1, 4, 5, 8,10,12,14,15,17,19)( 2, 3, 6, 7, 9,11,13,16,18,20)$
10B3	$10^{2}$	$4$	$10$	$18$	$( 1, 8,14,19, 5,12,17, 4,10,15)( 2, 7,13,20, 6,11,18, 3, 9,16)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	2C	2D	2E	4A1	4A-1	4B1	4B-1	5A	10A	10B1	10B3
	Size	1	1	2	5	5	10	10	10	10	10	4	4	4	4
	2 P	1A	1A	1A	1A	1A	1A	2C	2C	2D	2D	5A	5A	5A	5A
	5 P	1A	2A	2B	2C	2D	2E	4A1	4A-1	4B1	4B-1	1A	2A	2B	2B
	Type
80.34.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.34.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
80.34.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
80.34.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
80.34.1e1	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-i$	$i$	$-i$	$i$	$1$	$1$	$-1$	$-1$
80.34.1e2	C	$1$	$1$	$-1$	$-1$	$-1$	$1$	$i$	$-i$	$i$	$-i$	$1$	$1$	$-1$	$-1$
80.34.1f1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$1$	$1$
80.34.1f2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$1$	$1$
80.34.2a	R	$2$	$-2$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$
80.34.2b	R	$2$	$-2$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$
80.34.4a	R	$4$	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
80.34.4b	R	$4$	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$	$1$
80.34.4c1	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$2{\zeta }_5^{-2}+1+2{\zeta }_5^2$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5^2$
80.34.4c2	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5^2$	$2{\zeta }_5^{-2}+1+2{\zeta }_5^2$

Regular extensions

Data not computed