Galois group: 20T22

• Label: 20T22
• Degree: $20$
• Order: $80$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-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 $t$:		$22$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,14,5,12)(2,13,6,11)(3,17,4,18)(7,15,9,19)(8,16,10,20)$, $(1,5,10,3,7,2,6,9,4,8)(11,15,20,14,18)(12,16,19,13,17)$

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: $F_5$

Low degree siblings

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

Malle's constant $a(G)$:     $1/5$

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