Galois group: 26T47

• Label: 26T47
• Degree: $26$
• Order: $11232$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\GL(3,3)$

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$47$
Group:		$\GL(3,3)$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,14,6,17,12,7,22,20)(2,13,5,18,11,8,21,19)(3,24,16,25,4,23,15,26)(9,10), (1,3,26,7,16,13,22,5)(2,4,25,8,15,14,21,6)(9,17,20,24,10,18,19,23)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$5616$:	$\PSL(3,3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 13: $\PSL(3,3)$

Low degree siblings

26T47, 26T48 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$117$	$2$	$( 1,21)( 2,22)( 3,15)( 4,16)( 5,12)( 6,11)( 7, 8)( 9,10)(13,14)(17,19)(18,20) (25,26)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$117$	$2$	$( 1,22)( 2,21)( 3,16)( 4,15)( 5,11)( 6,12)(17,20)(18,19)(23,24)$
	$ 4, 4, 4, 4, 4, 4, 1, 1 $	$702$	$4$	$( 1,20,21,18)( 2,19,22,17)( 3,12,15, 5)( 4,11,16, 6)( 7,26, 8,25)( 9,13,10,14)$
	$ 4, 4, 4, 4, 4, 4, 2 $	$702$	$4$	$( 1,19,21,17)( 2,20,22,18)( 3,11,15, 6)( 4,12,16, 5)( 7,25, 8,26)( 9,14,10,13) (23,24)$
	$ 8, 8, 8, 2 $	$702$	$8$	$( 1,12,20,15,21, 5,18, 3)( 2,11,19,16,22, 6,17, 4)( 7, 9,26,13, 8,10,25,14) (23,24)$
	$ 8, 8, 8, 1, 1 $	$702$	$8$	$( 1,11,20,16,21, 6,18, 4)( 2,12,19,15,22, 5,17, 3)( 7,10,26,14, 8, 9,25,13)$
	$ 8, 8, 8, 1, 1 $	$702$	$8$	$( 1, 4,18, 6,21,16,20,11)( 2, 3,17, 5,22,15,19,12)( 7,13,25, 9, 8,14,26,10)$
	$ 8, 8, 8, 2 $	$702$	$8$	$( 1, 3,18, 5,21,15,20,12)( 2, 4,17, 6,22,16,19,11)( 7,14,25,10, 8,13,26, 9) (23,24)$
	$ 13, 13 $	$432$	$13$	$( 1, 4,25, 6,10, 8,24,17,21,13,12,19,16)( 2, 3,26, 5, 9, 7,23,18,22,14,11,20, 15)$
	$ 26 $	$432$	$26$	$( 1, 3,25, 5,10, 7,24,18,21,14,12,20,16, 2, 4,26, 6, 9, 8,23,17,22,13,11,19,15 )$
	$ 26 $	$432$	$26$	$( 1,22, 6,20,24, 3,13, 9,16,18,25,11, 8, 2,21, 5,19,23, 4,14,10,15,17,26,12, 7 )$
	$ 13, 13 $	$432$	$13$	$( 1,21, 6,19,24, 4,13,10,16,17,25,12, 8)( 2,22, 5,20,23, 3,14, 9,15,18,26,11, 7)$
	$ 13, 13 $	$432$	$13$	$( 1,16,19,12,13,21,17,24, 8,10, 6,25, 4)( 2,15,20,11,14,22,18,23, 7, 9, 5,26, 3)$
	$ 26 $	$432$	$26$	$( 1,15,19,11,13,22,17,23, 8, 9, 6,26, 4, 2,16,20,12,14,21,18,24, 7,10, 5,25, 3 )$
	$ 26 $	$432$	$26$	$( 1, 7,12,26,17,15,10,14, 4,23,19, 5,21, 2, 8,11,25,18,16, 9,13, 3,24,20, 6,22 )$
	$ 13, 13 $	$432$	$13$	$( 1, 8,12,25,17,16,10,13, 4,24,19, 6,21)( 2, 7,11,26,18,15, 9,14, 3,23,20, 5, 22)$
	$ 6, 6, 6, 2, 2, 2, 2 $	$104$	$6$	$( 1,18,25, 2,17,26)( 3,22, 9, 4,21,10)( 5, 6)( 7, 8)(11,12)(13,19,15,14,20,16) (23,24)$
	$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $	$104$	$3$	$( 1,17,25)( 2,18,26)( 3,21, 9)( 4,22,10)(13,20,15)(14,19,16)$
	$ 6, 6, 3, 3, 2, 2, 2, 1, 1 $	$936$	$6$	$( 1, 4,17,22,25,10)( 2, 3,18,21,26, 9)( 7,11)( 8,12)(13,15,20)(14,16,19) (23,24)$
	$ 6, 6, 6, 2, 2, 2, 1, 1 $	$936$	$6$	$( 1, 3,17,21,25, 9)( 2, 4,18,22,26,10)( 5, 6)( 7,12)( 8,11)(13,16,20,14,15,19)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $	$624$	$3$	$( 1,16,11)( 2,15,12)( 3,20,14)( 4,19,13)( 5,18, 8)( 6,17, 7)(21,23,26) (22,24,25)$
	$ 6, 6, 6, 6, 2 $	$624$	$6$	$( 1,15,11, 2,16,12)( 3,19,14, 4,20,13)( 5,17, 8, 6,18, 7)( 9,10) (21,24,26,22,23,25)$

Group invariants

Order:		$11232=2^{5} \cdot 3^{3} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		11232.a
Character table:

	Size
	2 P
	3 P
	13 P
	Type