Galois group: 10T11

• Label: 10T11
• Degree: $10$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $A_5\times C_2$

Group invariants

Abstract group:		$A_5\times C_2$
Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$11$
CHM label:		$A(5)[x]2$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(2,4,10)(5,7,9)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$, $(1,3,5,7,9)(2,4,6,8,10)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$60$:	$A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $A_5$

Low degree siblings

12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61

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	$2^{5}$	$1$	$2$	$5$	$( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B	$2^{5}$	$15$	$2$	$5$	$( 1, 6)( 2, 9)( 3,10)( 4, 7)( 5, 8)$
2C	$2^{4},1^{2}$	$15$	$2$	$4$	$(1,7)(2,6)(3,9)(4,8)$
3A	$3^{2},1^{4}$	$20$	$3$	$4$	$( 2,10, 4)( 5, 9, 7)$
5A1	$5^{2}$	$12$	$5$	$8$	$( 1, 7, 3, 5, 9)( 2, 8,10, 4, 6)$
5A2	$5^{2}$	$12$	$5$	$8$	$( 1, 3, 9, 7, 5)( 2,10, 6, 8, 4)$
6A	$6,2^{2}$	$20$	$6$	$7$	$( 1, 6)( 2, 9,10, 7, 4, 5)( 3, 8)$
10A1	$10$	$12$	$10$	$9$	$( 1,10, 7, 4, 3, 6, 5, 2, 9, 8)$
10A3	$10$	$12$	$10$	$9$	$( 1, 4, 5, 8, 7, 6, 9,10, 3, 2)$

Malle's constant $a(G)$:     $1/4$

Character table

		1A	2A	2B	2C	3A	5A1	5A2	6A	10A1	10A3
	Size	1	1	15	15	20	12	12	20	12	12
	2 P	1A	1A	1A	1A	3A	5A2	5A1	3A	5A1	5A2
	3 P	1A	2A	2B	2C	1A	5A2	5A1	2A	10A3	10A1
	5 P	1A	2A	2B	2C	3A	1A	1A	6A	2A	2A
	Type
120.35.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.35.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
120.35.3a1	R	$3$	$3$	$-1$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
120.35.3a2	R	$3$	$3$	$-1$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
120.35.3b1	R	$3$	$-3$	$1$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$
120.35.3b2	R	$3$	$-3$	$1$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$
120.35.4a	R	$4$	$4$	$0$	$0$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
120.35.4b	R	$4$	$-4$	$0$	$0$	$1$	$-1$	$-1$	$-1$	$1$	$1$
120.35.5a	R	$5$	$5$	$1$	$1$	$-1$	$0$	$0$	$-1$	$0$	$0$
120.35.5b	R	$5$	$-5$	$-1$	$1$	$-1$	$0$	$0$	$1$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$108 x^{10} + \left(125 t^{2} - 25\right) x^{4} + \left(-50 t^{2} + 10\right) x^{2} + \left(5 t^{2} - 1\right)$