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

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