Galois group: 26T4

• Label: 26T4
• Degree: $26$
• Order: $52$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_{13}:C_4$

Group invariants

Abstract group:		$C_{13}:C_4$
Order:		$52=2^{2} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $C_{13}:C_4$

Low degree siblings

13T4

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	Index	Representative
1A	$1^{26}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{2}$	$13$	$2$	$12$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,25)(10,26)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20)$
4A1	$4^{6},2$	$13$	$4$	$19$	$( 1,19, 7,15)( 2,20, 8,16)( 3,10, 5,26)( 4, 9, 6,25)(11,22,23,14)(12,21,24,13)(17,18)$
4A-1	$4^{6},2$	$13$	$4$	$19$	$( 1,15, 7,19)( 2,16, 8,20)( 3,26, 5,10)( 4,25, 6, 9)(11,14,23,22)(12,13,24,21)(17,18)$
13A1	$13^{2}$	$4$	$13$	$24$	$( 1,25,24,22,20,18,16,14,12, 9, 7, 5, 3)( 2,26,23,21,19,17,15,13,11,10, 8, 6, 4)$
13A2	$13^{2}$	$4$	$13$	$24$	$( 1,24,20,16,12, 7, 3,25,22,18,14, 9, 5)( 2,23,19,15,11, 8, 4,26,21,17,13,10, 6)$
13A4	$13^{2}$	$4$	$13$	$24$	$( 1,20,12, 3,22,14, 5,24,16, 7,25,18, 9)( 2,19,11, 4,21,13, 6,23,15, 8,26,17,10)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	4A1	4A-1	13A1	13A2	13A4
	Size	1	13	13	13	4	4	4
	2 P	1A	1A	2A	2A	13A2	13A4	13A1
	13 P	1A	2A	4A-1	4A1	13A2	13A4	13A1
	Type
52.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
52.3.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
52.3.1c1	C	$1$	$-1$	$-i$	$i$	$1$	$1$	$1$
52.3.1c2	C	$1$	$-1$	$i$	$-i$	$1$	$1$	$1$
52.3.4a1	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$
52.3.4a2	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$
52.3.4a3	R	$4$	$0$	$0$	$0$	${\zeta }_{13}^{-3}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^3$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-4}+{\zeta }_{13}^4+{\zeta }_{13}^6$	${\zeta }_{13}^{-5}+{\zeta }_{13}^{-1}+{\zeta }_{13}+{\zeta }_{13}^5$

Regular extensions

Data not computed