Galois group: 26T21

• Label: 26T21
• Degree: $26$
• Order: $2028$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{13}:F_{13}$

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$21$
Group:		$C_{13}:F_{13}$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18), (1,4,3,12,9,10)(2,8,6,11,5,7)(14,26,22,19,20,24)(15,17,25,18,16,21)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$4$:	$C_4$
$6$:	$C_6$
$12$:	$C_{12}$
$156$:	$F_{13}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T21 x 5

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 13, 13 $	$12$	$13$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
	$ 13, 13 $	$12$	$13$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
	$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$12$	$13$	$(14,20,26,19,25,18,24,17,23,16,22,15,21)$
	$ 13, 13 $	$12$	$13$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
	$ 13, 13 $	$12$	$13$	$( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
	$ 13, 13 $	$12$	$13$	$( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
	$ 13, 13 $	$12$	$13$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
	$ 13, 13 $	$12$	$13$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
	$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$12$	$13$	$( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$
	$ 13, 13 $	$12$	$13$	$( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
	$ 13, 13 $	$12$	$13$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
	$ 13, 13 $	$12$	$13$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
	$ 13, 13 $	$12$	$13$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
	$ 13, 13 $	$12$	$13$	$( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$169$	$2$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
	$ 4, 4, 4, 4, 4, 4, 2 $	$169$	$4$	$( 1,15, 2,21)( 3,14,13,22)( 4,20,12,16)( 5,26,11,23)( 6,19,10,17)( 7,25, 9,24) ( 8,18)$
	$ 4, 4, 4, 4, 4, 4, 2 $	$169$	$4$	$( 1,26, 4,21)( 2,20, 3,14)( 5,15,13,19)( 6,22,12,25)( 7,16,11,18)( 8,23,10,24) ( 9,17)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $	$169$	$3$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
	$ 6, 6, 6, 6, 1, 1 $	$169$	$6$	$( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
	$ 12, 12, 2 $	$169$	$12$	$( 1,15, 6,14,13,23, 2,20,10,21, 3,25)( 4,17, 5,22, 9,16,12,18,11,26, 7,19) ( 8,24)$
	$ 12, 12, 2 $	$169$	$12$	$( 1,26, 8,17,10,20, 5,19,11,15, 9,25)( 2,21,12,23,13,18, 4,24, 7,22, 6,14) ( 3,16)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $	$169$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
	$ 6, 6, 6, 6, 1, 1 $	$169$	$6$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
	$ 12, 12, 2 $	$169$	$12$	$( 1,15, 5,23, 6,25, 3,19,12,24,11,22)( 2,17)( 4,21, 9,18, 7,14,13,26, 8,16,10, 20)$
	$ 12, 12, 2 $	$169$	$12$	$( 1,26, 7,14, 2,24, 4,20,11,19, 3,22)( 5,18, 8,25,12,17,13,15,10,21, 6,16) ( 9,23)$

Group invariants

Order:		$2028=2^{2} \cdot 3 \cdot 13^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		2028.q
Character table:

	Size
	2 P
	3 P
	13 P
	Type