Galois group: 20T27

• Label: 20T27
• Degree: $20$
• Order: $100$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_5:F_5$

Group invariants

Abstract group:		$C_5:F_5$
Order:		$100=2^{2} \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$27$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$10$
Generators:		$(1,12,18,7)(2,11,17,8)(3,5,15,14)(4,6,16,13)(9,20,10,19)$, $(3,8,12,16,19)(4,7,11,15,20)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$20$:	$F_5$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

Low degree siblings

10T10 x 2, 20T27, 25T10

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	$2^{10}$	$25$	$2$	$10$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,10)( 6, 9)( 7,12)( 8,11)(17,18)(19,20)$
4A1	$4^{5}$	$25$	$4$	$15$	$( 1, 3,13,15)( 2, 4,14,16)( 5, 7,10,12)( 6, 8, 9,11)(17,19,18,20)$
4A-1	$4^{5}$	$25$	$4$	$15$	$( 1,15,13, 3)( 2,16,14, 4)( 5,12,10, 7)( 6,11, 9, 8)(17,20,18,19)$
5A	$5^{4}$	$4$	$5$	$16$	$( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3, 8,12,16,19)( 4, 7,11,15,20)$
5B	$5^{4}$	$4$	$5$	$16$	$( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$
5C1	$5^{4}$	$4$	$5$	$16$	$( 1,17,14,10, 6)( 2,18,13, 9, 5)( 3,19,16,12, 8)( 4,20,15,11, 7)$
5C2	$5^{4}$	$4$	$5$	$16$	$( 1,14, 6,17,10)( 2,13, 5,18, 9)( 3,16, 8,19,12)( 4,15, 7,20,11)$
5D1	$5^{2},1^{10}$	$4$	$5$	$8$	$( 1,10,17, 6,14)( 2, 9,18, 5,13)$
5D2	$5^{2},1^{10}$	$4$	$5$	$8$	$( 3, 8,12,16,19)( 4, 7,11,15,20)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	4A1	4A-1	5A	5B	5C1	5C2	5D1	5D2
	Size	1	25	25	25	4	4	4	4	4	4
	2 P	1A	1A	2A	2A	5A	5B	5C2	5C1	5D2	5D1
	5 P	1A	2A	4A1	4A-1	1A	1A	1A	1A	1A	1A
	Type
100.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$	$1$
100.12.4a	R	$4$	$0$	$0$	$0$	$-1$	$-1$	$4$	$-1$	$-1$	$-1$
100.12.4b	R	$4$	$0$	$0$	$0$	$4$	$-1$	$-1$	$-1$	$-1$	$-1$
100.12.4c1	R	$4$	$0$	$0$	$0$	$-1$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$-1$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$
100.12.4c2	R	$4$	$0$	$0$	$0$	$-1$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$-1$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$
100.12.4d1	R	$4$	$0$	$0$	$0$	$-1$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$-1$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$
100.12.4d2	R	$4$	$0$	$0$	$0$	$-1$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	$-1$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	${\zeta }_5^{-2}+2+{\zeta }_5^2$

Regular extensions

Data not computed