Galois group: 12T179

• Label: 12T179
• Degree: $12$
• Order: $660$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: yes
• $p$-group: no
• Group: $\PSL(2,11)$

Group action invariants

Degree $n$:		$12$
Transitive number $t$:		$179$
Group:		$\PSL(2,11)$
CHM label:		$L(2,11)$
Parity:		$1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,12), (1,10)(2,5)(3,7)(4,8)(6,9)(11,12)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Low degree siblings

11T5 x 2

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 $	$1$	$1$	$()$
	$ 5, 5, 1, 1 $	$132$	$5$	$( 3, 4,10,12, 7)( 5,11, 9, 6, 8)$
	$ 5, 5, 1, 1 $	$132$	$5$	$( 3,10, 7, 4,12)( 5, 9, 8,11, 6)$
	$ 11, 1 $	$60$	$11$	$( 2, 3,11,10,12, 4, 6, 5, 8, 7, 9)$
	$ 11, 1 $	$60$	$11$	$( 2, 5,10, 9, 6,11, 7, 4, 3, 8,12)$
	$ 2, 2, 2, 2, 2, 2 $	$55$	$2$	$( 1, 2)( 3, 5)( 4, 8)( 6,10)( 7,11)( 9,12)$
	$ 3, 3, 3, 3 $	$110$	$3$	$( 1, 2, 3)( 4, 8,10)( 5, 7,12)( 6,11, 9)$
	$ 6, 6 $	$110$	$6$	$( 1, 2, 3, 5,11,10)( 4,12, 8, 6, 9, 7)$

Group invariants

Order:		$660=2^{2} \cdot 3 \cdot 5 \cdot 11$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		660.13
Character table:

		1A	2A	3A	5A1	5A2	6A	11A1	11A-1
	Size	1	55	110	132	132	110	60	60
	2 P	1A	1A	3A	5A2	5A1	3A	11A-1	11A1
	3 P	1A	2A	1A	5A2	5A1	2A	11A1	11A-1
	5 P	1A	2A	3A	1A	1A	6A	11A1	11A-1
	11 P	1A	2A	3A	5A1	5A2	6A	1A	1A
	Type
660.13.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
660.13.5a1	C	$5$	$1$	$-1$	$0$	$0$	$1$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$
660.13.5a2	C	$5$	$1$	$-1$	$0$	$0$	$1$	${\zeta }_{11}^{-2}+{\zeta }_{11}+{\zeta }_{11}^3+{\zeta }_{11}^4+{\zeta }_{11}^5$	$-{\zeta }_{11}^{-2}-1-{\zeta }_{11}-{\zeta }_{11}^3-{\zeta }_{11}^4-{\zeta }_{11}^5$
660.13.10a	R	$10$	$2$	$1$	$0$	$0$	$-1$	$-1$	$-1$
660.13.10b	R	$10$	$-2$	$1$	$0$	$0$	$1$	$-1$	$-1$
660.13.11a	R	$11$	$-1$	$-1$	$1$	$1$	$-1$	$0$	$0$
660.13.12a1	R	$12$	$0$	$0$	${\zeta }_5^{-2}+{\zeta }_5^2$	${\zeta }_5^{-1}+{\zeta }_5$	$0$	$1$	$1$
660.13.12a2	R	$12$	$0$	$0$	${\zeta }_5^{-1}+{\zeta }_5$	${\zeta }_5^{-2}+{\zeta }_5^2$	$0$	$1$	$1$