Galois group: 20T23

• Label: 20T23
• Degree: $20$
• Order: $80$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^4:C_5$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$23$
Group:		$C_2^4:C_5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		(3,14)(4,13)(5,6)(7,8)(9,20)(10,19)(15,16)(17,18), (1,6,9,13,17)(2,5,10,14,18)(3,7,11,16,19)(4,8,12,15,20)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$5$:	$C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2^4 : C_5$ x 3

Low degree siblings

10T8 x 3, 16T178, 20T17 x 6, 40T57 x 3

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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$5$	$2$	$( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,10)(13,14)(19,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$5$	$2$	$( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,19)(10,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$5$	$2$	$( 3,14)( 4,13)( 5, 6)( 7, 8)( 9,20)(10,19)(15,16)(17,18)$
	$ 5, 5, 5, 5 $	$16$	$5$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$
	$ 5, 5, 5, 5 $	$16$	$5$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$
	$ 5, 5, 5, 5 $	$16$	$5$	$( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)(11,17,13,19,15)(12,18,14,20,16)$
	$ 5, 5, 5, 5 $	$16$	$5$	$( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)(11,19,17,15,13)(12,20,18,16,14)$

Group invariants

Order:		$80=2^{4} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		80.49
Character table:

		1A	2A	2B	2C	5A1	5A-1	5A2	5A-2
	Size	1	5	5	5	16	16	16	16
	2 P	1A	1A	1A	1A	5A-1	5A1	5A2	5A-2
	5 P	1A	2C	2A	2B	1A	1A	1A	1A
	Type
80.49.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.49.1b1	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
80.49.1b2	C	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
80.49.1b3	C	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
80.49.1b4	C	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$
80.49.5a	R	$5$	$-3$	$1$	$1$	$0$	$0$	$0$	$0$
80.49.5b	R	$5$	$1$	$-3$	$1$	$0$	$0$	$0$	$0$
80.49.5c	R	$5$	$1$	$1$	$-3$	$0$	$0$	$0$	$0$