Galois group: 26T34

• Label: 26T34
• Degree: $26$
• Order: $4056$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{13}^2:(C_3\times Q_8)$

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$34$
Group:		$C_{13}^2:(C_3\times Q_8)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(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), (1,6,9,3,2,4,13,8,5,11,12,10)(14,20,19,17,26,18,15,22,23,25,16,24)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$C_6$ x 3
$8$:	$Q_8$
$12$:	$C_6\times C_2$
$24$:	24T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T34 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
	$1^{26}$	$1$	$1$	$()$
	$13^{2}$	$24$	$13$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
	$13^{2}$	$24$	$13$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
	$13,1^{13}$	$24$	$13$	$(14,16,18,20,22,24,26,15,17,19,21,23,25)$
	$13^{2}$	$24$	$13$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
	$13^{2}$	$24$	$13$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
	$13^{2}$	$24$	$13$	$( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
	$13^{2}$	$24$	$13$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
	$3^{8},1^{2}$	$169$	$3$	$( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
	$3^{8},1^{2}$	$169$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
	$6^{4},1^{2}$	$169$	$6$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
	$2^{12},1^{2}$	$169$	$2$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
	$6^{4},1^{2}$	$169$	$6$	$( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
	$12^{2},2$	$338$	$12$	$( 1,26, 8,17,10,20, 5,19,11,15, 9,25)( 2,21,12,23,13,18, 4,24, 7,22, 6,14) ( 3,16)$
	$12^{2},2$	$338$	$12$	$( 1,24, 6,14, 4,18,10,19, 5,16, 7,25)( 2,22, 3,20,13,26, 9,21, 8,23,11,17) (12,15)$
	$4^{6},2$	$338$	$4$	$( 1,18, 2,25)( 3,19,13,24)( 4,26,12,17)( 5,20,11,23)( 6,14,10,16)( 7,21, 9,22) ( 8,15)$
	$12^{2},1^{2}$	$338$	$12$	$( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,16,18,22,17,20,26,25,23,19,24,21)$
	$12^{2},1^{2}$	$338$	$12$	$( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,20,24,22,23,16,26,21,17,19,18,25)$
	$4^{6},1^{2}$	$338$	$4$	$( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
	$12^{2},2$	$338$	$12$	$( 1,26)( 2,23,11,22,10,25,13,16, 4,17, 5,14)( 3,20, 8,18, 6,24,12,19, 7,21, 9, 15)$
	$4^{6},2$	$338$	$4$	$( 1,24, 8,26)( 2,15, 7,22)( 3,19, 6,18)( 4,23, 5,14)( 9,17,13,20)(10,21,12,16) (11,25)$
	$12^{2},2$	$338$	$12$	$( 1,18, 7,25, 5,14,10,22, 4,15, 6,26)( 2,17,11,21, 8,24, 9,23,13,19, 3,16) (12,20)$

Group invariants

Order:		$4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		4056.bf
Character table:

	Size
	2 P
	3 P
	13 P
	Type