Galois group: 12T33

• Label: 12T33
• Degree: $12$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $A_5$

Group invariants

Abstract group:		$A_5$
Order:		$60=2^{2} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$33$
CHM label:		$A_{5}(12)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,3,5,7,9)(2,4,6,8,12)$, $(1,11,5)(2,7,9)(3,6,8)(4,12,10)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $\PSL(2,5)$

Low degree siblings

5T4, 6T12, 10T7, 15T5, 20T15, 30T9

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	$1^{12}$	$1$	$1$	$0$	$()$
2A	$2^{6}$	$15$	$2$	$6$	$( 1,12)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(10,11)$
3A	$3^{4}$	$20$	$3$	$8$	$( 1, 4, 6)( 2, 8,10)( 3, 9,11)( 5, 7,12)$
5A1	$5^{2},1^{2}$	$12$	$5$	$8$	$( 1,10, 5, 8, 6)( 4, 9, 7,12,11)$
5A2	$5^{2},1^{2}$	$12$	$5$	$8$	$( 1, 5, 6,10, 8)( 4, 7,11, 9,12)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	3A	5A1	5A2
	Size	1	15	20	12	12
	2 P	1A	1A	3A	5A2	5A1
	3 P	1A	2A	1A	5A2	5A1
	5 P	1A	2A	3A	1A	1A
	Type
60.5.1a	R	$1$	$1$	$1$	$1$	$1$
60.5.3a1	R	$3$	$-1$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
60.5.3a2	R	$3$	$-1$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$
60.5.4a	R	$4$	$0$	$1$	$-1$	$-1$
60.5.5a	R	$5$	$1$	$-1$	$0$	$0$

Regular extensions

Data not computed