Galois group: 10T3

• Label: 10T3
• Degree: $10$
• Order: $20$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_{10}$

Group invariants

Abstract group:		$D_{10}$
Order:		$20=2^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$3$
CHM label:		$D_{10}(10)=[D(5)]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,2,3,4,5,6,7,8,9,10)$, $(1,8)(2,7)(3,6)(4,5)(9,10)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$10$:	$D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $D_{5}$

Low degree siblings

10T3, 20T4

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	$1^{10}$	$1$	$1$	$0$	$()$
2A	$2^{5}$	$1$	$2$	$5$	$( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B	$2^{4},1^{2}$	$5$	$2$	$4$	$( 2,10)( 3, 9)( 4, 8)( 5, 7)$
2C	$2^{5}$	$5$	$2$	$5$	$( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)$
5A1	$5^{2}$	$2$	$5$	$8$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A2	$5^{2}$	$2$	$5$	$8$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
10A1	$10$	$2$	$10$	$9$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
10A3	$10$	$2$	$10$	$9$	$( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	5A1	5A2	10A1	10A3
	Size	1	1	5	5	2	2	2	2
	2 P	1A	1A	1A	1A	5A2	5A1	5A1	5A2
	5 P	1A	2A	2B	2C	1A	1A	2A	2A
	Type
20.4.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
20.4.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
20.4.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
20.4.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
20.4.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
20.4.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
20.4.2b1	R	$2$	$-2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
20.4.2b2	R	$2$	$-2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$

Regular extensions

$f_{ 1 } =$

$x^{10} + \left(t + 10\right) x^{8} + \left(-20 t + 33\right) x^{6} + \left(132 t + 40\right) x^{4} + \left(-320 t + 16\right) x^{2} + 256 t$