Galois group: 20T26

• Label: 20T26
• Degree: $20$
• Order: $100$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Transitivity: $1$
• 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 $t$:		$26$
Parity:		$-1$
Transitivity:		1
Primitive:		no
$\card{\Aut(F/K)}$:		$5$
Generators:		$(1,15,20,10)(2,12,19,8)(3,14,18,6)(4,11,17,9)(5,13,16,7)$, $(1,19,5,20,4,16,3,17,2,18)(6,11,9,13,7,15,10,12,8,14)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$10$:	$D_{5}$
$20$:	$F_5$, 20T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

Low degree siblings

25T11

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	$2^{10}$	$5$	$2$	$10$	$( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,14)( 7,13)( 8,12)( 9,11)(10,15)$
4A1	$4^{5}$	$25$	$4$	$15$	$( 1,15,16, 6)( 2,12,20, 9)( 3,14,19, 7)( 4,11,18,10)( 5,13,17, 8)$
4A-1	$4^{5}$	$25$	$4$	$15$	$( 1, 6,16,15)( 2, 9,20,12)( 3, 7,19,14)( 4,10,18,11)( 5, 8,17,13)$
5A1	$5^{4}$	$2$	$5$	$16$	$( 1, 4, 2, 5, 3)( 6, 7, 8, 9,10)(11,15,14,13,12)(16,18,20,17,19)$
5A2	$5^{4}$	$2$	$5$	$16$	$( 1, 2, 3, 4, 5)( 6, 8,10, 7, 9)(11,14,12,15,13)(16,20,19,18,17)$
5B	$5^{4}$	$4$	$5$	$16$	$( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)$
5C1	$5^{3},1^{5}$	$4$	$5$	$12$	$( 1, 3, 5, 2, 4)( 6, 7, 8, 9,10)(16,20,19,18,17)$
5C-1	$5^{3},1^{5}$	$4$	$5$	$12$	$( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19)$
5C2	$5^{3},1^{5}$	$4$	$5$	$12$	$( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)$
5C-2	$5^{3},1^{5}$	$4$	$5$	$12$	$( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,13,15,12,14)$
10A1	$10^{2}$	$10$	$10$	$18$	$( 1,16, 4,18, 2,20, 5,17, 3,19)( 6,11, 7,15, 8,14, 9,13,10,12)$
10A3	$10^{2}$	$10$	$10$	$18$	$( 1,17, 2,16, 3,20, 4,19, 5,18)( 6,13, 8,11,10,14, 7,12, 9,15)$

Malle's constant $a(G)$:     $1/10$

Character table

		1A	2A	4A1	4A-1	5A1	5A2	5B	5C1	5C-1	5C2	5C-2	10A1	10A3
	Size	1	5	25	25	2	2	4	4	4	4	4	10	10
	2 P	1A	1A	2A	2A	5A2	5A1	5B	5C2	5C-2	5C-1	5C1	5A1	5A2
	5 P	1A	2A	4A1	4A-1	1A	1A	1A	1A	1A	1A	1A	2A	2A
	Type
100.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
100.10.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
100.10.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
100.10.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
100.10.2a1	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
100.10.2a2	R	$2$	$2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
100.10.2b1	S	$2$	$-2$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
100.10.2b2	S	$2$	$-2$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$2$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
100.10.4a	R	$4$	$0$	$0$	$0$	$4$	$4$	$-1$	$-1$	$-1$	$-1$	$-1$	$0$	$0$
100.10.4b1	C	$4$	$0$	$0$	$0$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	${\zeta }_5^{-2}-{\zeta }_5-{\zeta }_5^2$	${\zeta }_5^{-2}+1+2{\zeta }_5$	$-1$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5-{\zeta }_5^2$	$1+{\zeta }_5+2{\zeta }_5^2$	$0$	$0$
100.10.4b2	C	$4$	$0$	$0$	$0$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$1+{\zeta }_5+2{\zeta }_5^2$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5-{\zeta }_5^2$	$-1$	${\zeta }_5^{-2}+1+2{\zeta }_5$	${\zeta }_5^{-2}-{\zeta }_5-{\zeta }_5^2$	$0$	$0$
100.10.4b3	C	$4$	$0$	$0$	$0$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5-{\zeta }_5^2$	${\zeta }_5^{-2}-{\zeta }_5-{\zeta }_5^2$	$-1$	$1+{\zeta }_5+2{\zeta }_5^2$	${\zeta }_5^{-2}+1+2{\zeta }_5$	$0$	$0$
100.10.4b4	C	$4$	$0$	$0$	$0$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	${\zeta }_5^{-2}+1+2{\zeta }_5$	$1+{\zeta }_5+2{\zeta }_5^2$	$-1$	${\zeta }_5^{-2}-{\zeta }_5-{\zeta }_5^2$	$-2{\zeta }_5^{-2}-1-2{\zeta }_5-{\zeta }_5^2$	$0$	$0$

Regular extensions

Data not computed