Galois group: 20T13

• Label: 20T13
• Degree: $20$
• Order: $40$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2\times F_5$

Group invariants

Abstract group:		$C_2\times F_5$
Order:		$40=2^{3} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $F_5$

Degree 10: $F_5$, $F_{5}\times C_2$ x 2

Low degree siblings

10T5 x 2, 20T9, 40T14

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

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	5A	10A
	Size	1	1	5	5	5	5	5	5	4	4
	2 P	1A	1A	1A	1A	2C	2C	2C	2C	5A	5A
	5 P	1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	1A	2A
	Type
40.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
40.12.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
40.12.1c	R	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
40.12.1d	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
40.12.1e1	C	$1$	$-1$	$1$	$-1$	$-i$	$i$	$-i$	$i$	$1$	$-1$
40.12.1e2	C	$1$	$-1$	$1$	$-1$	$i$	$-i$	$i$	$-i$	$1$	$-1$
40.12.1f1	C	$1$	$1$	$-1$	$-1$	$-i$	$-i$	$i$	$i$	$1$	$1$
40.12.1f2	C	$1$	$1$	$-1$	$-1$	$i$	$i$	$-i$	$-i$	$1$	$1$
40.12.4a	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
40.12.4b	R	$4$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$

Regular extensions

Data not computed