Galois group: 6T12

• Label: 6T12
• Degree: $6$
• Order: $60$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $\PSL(2,5)$

Group invariants

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

Group action invariants

Degree $n$:		$6$
Transitive number $t$:		$12$
CHM label:		$L(6) = PSL(2,5) = A_{5}(6)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,4)(5,6)$, $(1,2,3,4,6)$

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Low degree siblings

5T4, 10T7, 12T33, 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	$1^{6}$	$1$	$1$	$0$	$()$
2A	$2^{2},1^{2}$	$15$	$2$	$2$	$(1,2)(3,6)$
3A	$3^{2}$	$20$	$3$	$4$	$(1,4,2)(3,6,5)$
5A1	$5,1$	$12$	$5$	$4$	$(1,4,5,2,6)$
5A2	$5,1$	$12$	$5$	$4$	$(1,5,6,4,2)$

Malle's constant $a(G)$:     $1/2$

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

$f_{ 1 } =$

$x^{6} + 2 x^{4} + t x^{3} + 5 x^{2} + 6 x + 1$