Galois group: 10T10

• Label: 10T10
• Degree: $10$
• Order: $100$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_5^2 : C_4$

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$10$
Group:		$C_5^2 : C_4$
CHM label:		$1/2[D(5)^{2}]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(2,4,6,8,10), (1,6,9,4)(2,3,8,7)(5,10)

Low degree resolvents

|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 5: None

Low degree siblings

10T10, 20T27 x 2, 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	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 1, 1 $	$25$	$2$	$( 3, 9)( 4,10)( 5, 7)( 6, 8)$
	$ 5, 1, 1, 1, 1, 1 $	$4$	$5$	$( 2, 4, 6, 8,10)$
	$ 5, 1, 1, 1, 1, 1 $	$4$	$5$	$( 2, 6,10, 4, 8)$
	$ 4, 4, 2 $	$25$	$4$	$( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$
	$ 4, 4, 2 $	$25$	$4$	$( 1, 2)( 3,10, 9, 4)( 5, 8, 7, 6)$
	$ 5, 5 $	$4$	$5$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
	$ 5, 5 $	$4$	$5$	$( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
	$ 5, 5 $	$4$	$5$	$( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$
	$ 5, 5 $	$4$	$5$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

Group invariants

Order:		$100=2^{2} \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		100.12
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	5C2	5B	5A	5D1	5D2	5C1
	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	$4$	$0$	$0$	$0$	$-1$	$4$	$-1$	$-1$	$-1$	$-1$
100.12.4b	R	$4$	$0$	$0$	$0$	$4$	$-1$	$-1$	$-1$	$-1$	$-1$
100.12.4c1	R	$4$	$0$	$0$	$0$	$-1$	$-1$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$
100.12.4c2	R	$4$	$0$	$0$	$0$	$-1$	$-1$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	${\zeta }_5^{-2}+2+{\zeta }_5^2$
100.12.4d1	R	$4$	$0$	$0$	$0$	$-1$	$-1$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$
100.12.4d2	R	$4$	$0$	$0$	$0$	$-1$	$-1$	$-{\zeta }_5^{-2}+1-{\zeta }_5^2$	${\zeta }_5^{-2}+2+{\zeta }_5^2$	$2{\zeta }_5^{-2}+2{\zeta }_5^2$	$2{\zeta }_5^{-1}+2{\zeta }_5$