Galois group: 10T16

• Label: 10T16
• Degree: $10$
• Order: $160$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $(C_2^4 : C_5) : C_2$

Group invariants

Abstract group:		$(C_2^4 : C_5) : C_2$
Order:		$160=2^{5} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$16$
CHM label:		$1/2[2^{5}]D(5)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,9)(2,8)(3,7)(4,6)(5,10)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $D_{5}$

Low degree siblings

10T15 x 3, 10T16 x 2, 16T415, 20T38 x 6, 20T39, 20T43 x 3, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

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	$2^{4},1^{2}$	$5$	$2$	$4$	$( 1, 6)( 2, 7)( 4, 9)( 5,10)$
2B	$2^{2},1^{6}$	$5$	$2$	$2$	$(2,7)(3,8)$
2C	$2^{2},1^{6}$	$5$	$2$	$2$	$( 2, 7)( 5,10)$
2D	$2^{5}$	$20$	$2$	$5$	$( 1, 7)( 2, 6)( 3,10)( 4, 9)( 5, 8)$
4A	$4^{2},2$	$20$	$4$	$7$	$( 1,10, 6, 5)( 2, 4, 7, 9)( 3, 8)$
4B	$4,2^{2},1^{2}$	$20$	$4$	$5$	$(1,9)(2,3,7,8)(4,6)$
4C	$4,2^{2},1^{2}$	$20$	$4$	$5$	$( 2, 5, 7,10)( 3, 9)( 4, 8)$
5A1	$5^{2}$	$32$	$5$	$8$	$( 1, 3,10, 7, 4)( 2, 9, 6, 8, 5)$
5A2	$5^{2}$	$32$	$5$	$8$	$( 1,10, 4, 3, 7)( 2, 6, 5, 9, 8)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	2D	4A	4B	4C	5A1	5A2
	Size	1	5	5	5	20	20	20	20	32	32
	2 P	1A	1A	1A	1A	1A	2A	2B	2C	5A2	5A1
	5 P	1A	2A	2B	2C	2D	4A	4B	4C	1A	1A
	Type
160.234.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
160.234.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
160.234.2a1	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
160.234.2a2	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
160.234.5a	R	$5$	$-3$	$1$	$1$	$1$	$1$	$-1$	$-1$	$0$	$0$
160.234.5b	R	$5$	$1$	$-3$	$1$	$1$	$-1$	$1$	$-1$	$0$	$0$
160.234.5c	R	$5$	$1$	$1$	$-3$	$1$	$-1$	$-1$	$1$	$0$	$0$
160.234.5d	R	$5$	$-3$	$1$	$1$	$-1$	$-1$	$1$	$1$	$0$	$0$
160.234.5e	R	$5$	$1$	$-3$	$1$	$-1$	$1$	$-1$	$1$	$0$	$0$
160.234.5f	R	$5$	$1$	$1$	$-3$	$-1$	$1$	$1$	$-1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$16 t x^{10} + \left(t^{2} + 50 t - 9375\right) x^{6} + \left(-3 t^{2} + 250 t + 28125\right) x^{4} + \left(4 t^{2} + 5000 t - 37500\right)$