Galois group: 20T5

• Label: 20T5
• Degree: $20$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $F_5$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$5$
Group:		$F_5$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$20$
Generators:		(1,4,17,15)(2,3,18,16)(5,12,14,7)(6,11,13,8)(9,19,10,20), (1,6,10,14,18)(2,5,9,13,17)(3,8,12,15,19)(4,7,11,16,20)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

5T3, 10T4

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

Group invariants

Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		20.3
Character table:

		1A	2A	4A1	4A-1	5A
	Size	1	5	5	5	4
	2 P	1A	1A	2A	2A	5A
	5 P	1A	2A	4A1	4A-1	1A
	Type
20.3.1a	R	$1$	$1$	$1$	$1$	$1$
20.3.1b	R	$1$	$1$	$-1$	$-1$	$1$
20.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$
20.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$
20.3.4a	R	$4$	$0$	$0$	$0$	$-1$