Galois group: 26T30

• Label: 26T30
• Degree: $26$
• Order: $4056$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_{13}^2:(C_4\times S_3)$

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$30$
Group:		$C_{13}^2:(C_4\times S_3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$
$24$:	$S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T42 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
	$1^{26}$	$1$	$1$	$()$
	$13^{2}$	$24$	$13$	$( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
	$13^{2}$	$24$	$13$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
	$13^{2}$	$24$	$13$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
	$13,1^{13}$	$24$	$13$	$(14,19,24,16,21,26,18,23,15,20,25,17,22)$
	$13^{2}$	$12$	$13$	$( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
	$13^{2}$	$12$	$13$	$( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
	$13^{2}$	$12$	$13$	$( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
	$13^{2}$	$12$	$13$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
	$13^{2}$	$12$	$13$	$( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
	$13^{2}$	$12$	$13$	$( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
	$2^{12},1^{2}$	$169$	$2$	$( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
	$3^{8},1^{2}$	$338$	$3$	$( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
	$6^{4},1^{2}$	$338$	$6$	$( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$
	$4^{6},2$	$507$	$4$	$( 1,17, 5,24)( 2,22, 4,19)( 3,14)( 6,16,13,25)( 7,21,12,20)( 8,26,11,15) ( 9,18,10,23)$
	$4^{6},2$	$507$	$4$	$( 1,24)( 2,19,13,16)( 3,14,12,21)( 4,22,11,26)( 5,17,10,18)( 6,25, 9,23) ( 7,20, 8,15)$
	$12^{2},1^{2}$	$338$	$12$	$( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,25,18,19,17,21,26,16,23,22,24,20)$
	$12^{2},1^{2}$	$338$	$12$	$( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,16,18,22,17,20,26,25,23,19,24,21)$
	$4^{6},1^{2}$	$169$	$4$	$( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
	$4^{6},1^{2}$	$169$	$4$	$( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
	$26$	$156$	$26$	$( 1,17,12,25,10,20, 8,15, 6,23, 4,18, 2,26,13,21,11,16, 9,24, 7,19, 5,14, 3,22 )$
	$2^{13}$	$39$	$2$	$( 1,25)( 2,21)( 3,17)( 4,26)( 5,22)( 6,18)( 7,14)( 8,23)( 9,19)(10,15)(11,24) (12,20)(13,16)$
	$26$	$156$	$26$	$( 1,23, 7,25,13,14, 6,16,12,18, 5,20,11,22, 4,24,10,26, 3,15, 9,17, 2,19, 8,21 )$
	$26$	$156$	$26$	$( 1,16, 2,25, 3,21, 4,17, 5,26, 6,22, 7,18, 8,14, 9,23,10,19,11,15,12,24,13,20 )$
	$26$	$156$	$26$	$( 1,24, 8,26, 2,15, 9,17, 3,19,10,21, 4,23,11,25, 5,14,12,16, 6,18,13,20, 7,22 )$
	$2^{13}$	$39$	$2$	$( 1,19)( 2,23)( 3,14)( 4,18)( 5,22)( 6,26)( 7,17)( 8,21)( 9,25)(10,16)(11,20) (12,24)(13,15)$
	$26$	$156$	$26$	$( 1,17, 6,24,11,18, 3,25, 8,19,13,26, 5,20,10,14, 2,21, 7,15,12,22, 4,16, 9,23 )$
	$26$	$156$	$26$	$( 1,23, 4,22, 7,21,10,20,13,19, 3,18, 6,17, 9,16,12,15, 2,14, 5,26, 8,25,11,24 )$

Group invariants

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

	Size
	2 P
	3 P
	13 P
	Type