Galois group: 10T15

• Label: 10T15
• 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 $t$:		$15$
CHM label:		$[2^{4}]D(5)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,9)(2,8)(3,7)(4,6)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$

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 2, 10T16 x 3, 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	$2^{2},1^{6}$	$5$	$2$	$2$	$( 1, 6)( 5,10)$
2B	$2^{2},1^{6}$	$5$	$2$	$2$	$(1,6)(4,9)$
2C	$2^{4},1^{2}$	$5$	$2$	$4$	$( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2D	$2^{4},1^{2}$	$20$	$2$	$4$	$( 1, 2)( 3, 5)( 6, 7)( 8,10)$
4A	$4,2^{3}$	$20$	$4$	$6$	$( 1,10, 6, 5)( 2, 9)( 3, 8)( 4, 7)$
4B	$4,2^{3}$	$20$	$4$	$6$	$( 1, 4, 6, 9)( 2, 3)( 5,10)( 7, 8)$
4C	$4^{2},1^{2}$	$20$	$4$	$6$	$( 2,10, 7, 5)( 3, 4, 8, 9)$
5A1	$5^{2}$	$32$	$5$	$8$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A2	$5^{2}$	$32$	$5$	$8$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

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

$f_{ 1 } =$

$x^{10} + \left(t - 5\right) x^{6} - t x^{4} + 10 x^{2} - 4$