Galois group: 5T3

• Label: 5T3
• Degree: $5$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $F_5$

Group invariants

Abstract group:		$F_5$
Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$5$
Transitive number $t$:		$3$
CHM label:		$F(5) = 5:4$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,2,3,4,5)$, $(1,2,4,3)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

10T4, 20T5

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	$1^{5}$	$1$	$1$	$0$	$()$
2A	$2^{2},1$	$5$	$2$	$2$	$(1,4)(2,3)$
4A1	$4,1$	$5$	$4$	$3$	$(1,2,4,3)$
4A-1	$4,1$	$5$	$4$	$3$	$(1,3,4,2)$
5A	$5$	$4$	$5$	$4$	$(1,2,3,4,5)$

Malle's constant $a(G)$:     $1/2$

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	$4$	$0$	$0$	$0$	$-1$

Regular extensions

$f_{ 1 } =$	$x^{5} + \left(\left(s^{2} + 4\right) t + \left(-2 s - 17/4\right)\right) x^{4} + \left(\left(3 s^{2} + 12\right) t + \left(s^{2} + 13/2 s + 5\right)\right) x^{3} + \left(\left(-s^{2} - 4\right) t + \left(-11/2 s + 8\right)\right) x^{2} + \left(s - 6\right) x + 1$
	The polynomial $f_{1}$ is generic for any base field $K$ of characteristic $\neq$ 2
$f_{ 2 } =$	$x^{5}+10 x^{3}+5 t x^{2}-15 x+t^{2}-t+16$