Galois group: 20T17

• Label: 20T17
• Degree: $20$
• Order: $80$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^4:C_5$

Group invariants

Abstract group:		$C_2^4:C_5$
Order:		$80=2^{4} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$5$:	$C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$ x 2

Low degree siblings

10T8 x 3, 16T178, 20T17 x 5, 20T23, 40T57 x 3

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	$2^{6},1^{8}$	$5$	$2$	$6$	$( 5,16)( 6,15)( 7,17)( 8,18)( 9,10)(19,20)$
2B	$2^{10}$	$5$	$2$	$10$	$( 1,11)( 2,12)( 3, 4)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)(13,14)$
2C	$2^{8},1^{4}$	$5$	$2$	$8$	$( 3,14)( 4,13)( 5, 6)( 7,18)( 8,17)( 9,10)(15,16)(19,20)$
5A1	$5^{4}$	$16$	$5$	$16$	$( 1,14, 6,18,10)( 2,13, 5,17, 9)( 3,15, 7,19,12)( 4,16, 8,20,11)$
5A-1	$5^{4}$	$16$	$5$	$16$	$( 1,10,18, 6,14)( 2, 9,17, 5,13)( 3,12,19, 7,15)( 4,11,20, 8,16)$
5A2	$5^{4}$	$16$	$5$	$16$	$( 1, 6,10,14,18)( 2, 5, 9,13,17)( 3, 7,12,15,19)( 4, 8,11,16,20)$
5A-2	$5^{4}$	$16$	$5$	$16$	$( 1,18,14,10, 6)( 2,17,13, 9, 5)( 3,19,15,12, 7)( 4,20,16,11, 8)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	5A1	5A-1	5A2	5A-2
	Size	1	5	5	5	16	16	16	16
	2 P	1A	1A	1A	1A	5A2	5A-2	5A-1	5A1
	5 P	1A	2A	2B	2C	1A	1A	1A	1A
	Type
80.49.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.49.1b1	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
80.49.1b2	C	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
80.49.1b3	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
80.49.1b4	C	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$
80.49.5a	R	$5$	$-3$	$1$	$1$	$0$	$0$	$0$	$0$
80.49.5b	R	$5$	$1$	$-3$	$1$	$0$	$0$	$0$	$0$
80.49.5c	R	$5$	$1$	$1$	$-3$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed