Galois group: 20T62

• Label: 20T62
• Degree: $20$
• Order: $240$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $C_2\times S_5$

Group invariants

Abstract group:		$C_2\times S_5$
Order:		$240=2^{4} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$62$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,15,20,14)(2,16,19,13)(3,12)(4,11)(5,8,9,17)(6,7,10,18)$, $(1,7,4,2,8,3)(5,19,14,6,20,13)(9,10)(11,18,15,12,17,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$120$:	$S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

10T22 x 2, 12T123 x 2, 20T62, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 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	$2^{10}$	$10$	$2$	$10$	$( 1, 2)( 3, 9)( 4,10)( 5,12)( 6,11)( 7, 8)(13,14)(15,19)(16,20)(17,18)$
2C	$2^{6},1^{8}$	$10$	$2$	$6$	$( 1, 8)( 2, 7)( 5,20)( 6,19)(11,15)(12,16)$
2D	$2^{10}$	$15$	$2$	$10$	$( 1,12)( 2,11)( 3,17)( 4,18)( 5,19)( 6,20)( 7,15)( 8,16)( 9,10)(13,14)$
2E	$2^{8},1^{4}$	$15$	$2$	$8$	$( 3,19)( 4,20)( 5,17)( 6,18)( 7,16)( 8,15)( 9,14)(10,13)$
3A	$3^{6},1^{2}$	$20$	$3$	$12$	$( 1, 8,14)( 2, 7,13)( 3, 6,19)( 4, 5,20)( 9,11,15)(10,12,16)$
4A	$4^{4},2^{2}$	$30$	$4$	$14$	$( 1,11)( 2,12)( 3,18,19, 6)( 4,17,20, 5)( 7,10,16,13)( 8, 9,15,14)$
4B	$4^{4},2^{2}$	$30$	$4$	$14$	$( 1, 7, 5,19)( 2, 8, 6,20)( 3,14)( 4,13)( 9,12,17,16)(10,11,18,15)$
5A	$5^{4}$	$24$	$5$	$16$	$( 1,11, 4,15, 5)( 2,12, 3,16, 6)( 7,10,18,19,13)( 8, 9,17,20,14)$
6A	$6^{3},2$	$20$	$6$	$16$	$( 1,13, 8, 2,14, 7)( 3,15, 6, 9,19,11)( 4,16, 5,10,20,12)(17,18)$
6B	$6^{3},2$	$20$	$6$	$16$	$( 1,13,15, 2,14,16)( 3, 5,18, 4, 6,17)( 7,11,10, 8,12, 9)(19,20)$
6C	$6^{2},3^{2},1^{2}$	$20$	$6$	$14$	$( 1,11,20, 8,15, 5)( 2,12,19, 7,16, 6)( 3,10,18)( 4, 9,17)$
10A	$10^{2}$	$24$	$10$	$18$	$( 1,16,11, 6, 4, 2,15,12, 5, 3)( 7,20,10,14,18, 8,19, 9,13,17)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	2B	2C	2D	2E	3A	4A	4B	5A	6A	6B	6C	10A
	Size	1	1	10	10	15	15	20	30	30	24	20	20	20	24
	2 P	1A	1A	1A	1A	1A	1A	3A	2E	2E	5A	3A	3A	3A	5A
	3 P	1A	2A	2B	2C	2D	2E	1A	4A	4B	5A	2B	2A	2C	10A
	5 P	1A	2A	2B	2C	2D	2E	3A	4A	4B	1A	6A	6B	6C	2A
	Type
240.189.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
240.189.1b	R	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$
240.189.1c	R	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
240.189.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$
240.189.4a	R	$4$	$4$	$2$	$2$	$0$	$0$	$1$	$0$	$0$	$-1$	$-1$	$1$	$-1$	$-1$
240.189.4b	R	$4$	$4$	$-2$	$-2$	$0$	$0$	$1$	$0$	$0$	$-1$	$1$	$1$	$1$	$-1$
240.189.4c	R	$4$	$-4$	$-2$	$2$	$0$	$0$	$1$	$0$	$0$	$-1$	$1$	$-1$	$-1$	$1$
240.189.4d	R	$4$	$-4$	$2$	$-2$	$0$	$0$	$1$	$0$	$0$	$-1$	$-1$	$-1$	$1$	$1$
240.189.5a	R	$5$	$5$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$-1$	$0$
240.189.5b	R	$5$	$5$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$0$	$1$	$-1$	$1$	$0$
240.189.5c	R	$5$	$-5$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$0$	$-1$	$1$	$1$	$0$
240.189.5d	R	$5$	$-5$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$0$	$1$	$1$	$-1$	$0$
240.189.6a	R	$6$	$6$	$0$	$0$	$-2$	$-2$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$1$
240.189.6b	R	$6$	$-6$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$-1$

Regular extensions

Data not computed