Galois group: 26T49

• Label: 26T49
• Degree: $26$
• Order: $11232$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\SL(3,3):C_2$

Group invariants

Abstract group:		$\SL(3,3):C_2$
Order:		$11232=2^{5} \cdot 3^{3} \cdot 13$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$26$
Transitive number $t$:		$49$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,22,8,25,5,23)(2,26,13,21,4,16)(3,24,9,17,11,19)(6,18)(7,20,10,14,12,15)$, $(1,24,12,23,3,17,10,25,13,21,5,18)(2,20)(4,14,8,16,11,26)(6,15,7,19)(9,22)$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

There are no siblings 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	$2^{8},1^{10}$	$117$	$2$	$8$	$( 1, 3)( 4,11)( 5,10)( 6, 7)(14,16)(17,18)(21,25)(23,24)$
2B	$2^{13}$	$234$	$2$	$13$	$( 1,15)( 2,17)( 3,16)( 4,23)( 5,25)( 6,21)( 7,22)( 8,26)( 9,24)(10,14)(11,19)(12,20)(13,18)$
3A	$3^{6},1^{8}$	$104$	$3$	$12$	$( 2, 9, 6)( 3, 8,10)( 4,13, 5)(14,20,25)(15,18,16)(19,23,26)$
3B	$3^{8},1^{2}$	$624$	$3$	$16$	$( 1, 8, 7)( 2, 9,13)( 3,12, 6)( 5,10,11)(14,19,25)(15,26,22)(16,20,21)(17,24,18)$
4A	$4^{4},2^{5}$	$234$	$4$	$17$	$( 1,21)( 2,20,10,23)( 3,26, 9,25)( 4,16)( 5,18)( 6,14, 8,19)( 7,24)(11,17,12,22)(13,15)$
4B	$4^{4},2^{4},1^{2}$	$702$	$4$	$16$	$( 1, 4, 3,11)( 2,12)( 5, 7,10, 6)( 8,13)(14,24,16,23)(15,20)(17,25,18,21)(19,22)$
6A	$6^{2},3^{2},2^{2},1^{4}$	$936$	$6$	$16$	$( 2, 8, 9,10, 6, 3)( 4, 5,13)(11,12)(14,26,20,19,25,23)(15,16,18)(17,22)$
6B	$6^{4},2$	$1872$	$6$	$21$	$( 1,22, 8,15, 7,26)( 2,18, 9,17,13,24)( 3,21,12,16, 6,20)( 4,23)( 5,19,10,25,11,14)$
8A	$8^{2},4^{2},1^{2}$	$1404$	$8$	$20$	$( 1, 7, 4,10, 3, 6,11, 5)( 2, 8,12,13)(14,18,24,21,16,17,23,25)(15,22,20,19)$
8B	$8^{2},4^{2},2$	$1404$	$8$	$21$	$( 1,18, 6,23, 4,20,11,24)( 2,17,13,19)( 3,21, 9,14, 7,15, 5,22)( 8,25,12,16)(10,26)$
12A1	$12,6,4,2^{2}$	$936$	$12$	$21$	$( 1,21)( 2,26, 8,20, 9,19,10,25, 6,23, 3,14)( 4,15, 5,16,13,18)( 7,24)(11,22,12,17)$
12A5	$12,6,4,2^{2}$	$936$	$12$	$21$	$( 1,21)( 2,19, 3,20, 6,26,10,14, 9,23, 8,25)( 4,18,13,16, 5,15)( 7,24)(11,22,12,17)$
13A1	$13^{2}$	$864$	$13$	$24$	$( 1, 9, 7, 3, 6, 8, 4, 5,11,13, 2,10,12)(14,25,26,16,19,24,15,22,17,23,18,20,21)$
13A2	$13^{2}$	$864$	$13$	$24$	$( 1, 7, 6, 4,11, 2,12, 9, 3, 8, 5,13,10)(14,26,19,15,17,18,21,25,16,24,22,23,20)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	3A	3B	4A	4B	6A	6B	8A	8B	12A1	12A5	13A1	13A2
	Size	1	117	234	104	624	234	702	936	1872	1404	1404	936	936	864	864
	2 P	1A	1A	1A	3A	3B	2A	2A	3A	3B	4B	4B	6A	6A	13A2	13A1
	3 P	1A	2A	2B	1A	1A	4A	4B	2A	2B	8A	8B	4A	4A	13A1	13A2
	13 P	1A	2A	2B	3A	3B	4A	4B	6A	6B	8A	8B	12A1	12A5	1A	1A
	Type
11232.b.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
11232.b.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$
11232.b.12a1	R	$12$	$4$	$0$	$3$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-1$	$-1$
11232.b.12a2	R	$12$	$4$	$0$	$3$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	$-1$	$-1$
11232.b.13a	R	$13$	$-3$	$-1$	$4$	$1$	$3$	$1$	$0$	$-1$	$-1$	$1$	$0$	$0$	$0$	$0$
11232.b.13b	R	$13$	$-3$	$1$	$4$	$1$	$-3$	$1$	$0$	$1$	$-1$	$-1$	$0$	$0$	$0$	$0$
11232.b.26a	R	$26$	$2$	$-2$	$-1$	$-1$	$2$	$2$	$-1$	$1$	$0$	$0$	$-1$	$-1$	$0$	$0$
11232.b.26b	R	$26$	$2$	$2$	$-1$	$-1$	$-2$	$2$	$-1$	$-1$	$0$	$0$	$1$	$1$	$0$	$0$
11232.b.27a	R	$27$	$3$	$3$	$0$	$0$	$3$	$-1$	$0$	$0$	$-1$	$-1$	$0$	$0$	$1$	$1$
11232.b.27b	R	$27$	$3$	$-3$	$0$	$0$	$-3$	$-1$	$0$	$0$	$-1$	$1$	$0$	$0$	$1$	$1$
11232.b.32a1	R	$32$	$0$	$0$	$-4$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$	$-{\zeta }_{13}^{-6}-{\zeta }_{13}^{-5}-{\zeta }_{13}^{-2}-1-{\zeta }_{13}^2-{\zeta }_{13}^5-{\zeta }_{13}^6$
11232.b.32a2	R	$32$	$0$	$0$	$-4$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_{13}^{-6}-{\zeta }_{13}^{-5}-{\zeta }_{13}^{-2}-1-{\zeta }_{13}^2-{\zeta }_{13}^5-{\zeta }_{13}^6$	${\zeta }_{13}^{-6}+{\zeta }_{13}^{-5}+{\zeta }_{13}^{-2}+{\zeta }_{13}^2+{\zeta }_{13}^5+{\zeta }_{13}^6$
11232.b.39a	R	$39$	$-1$	$-3$	$3$	$0$	$1$	$-1$	$-1$	$0$	$1$	$-1$	$1$	$1$	$0$	$0$
11232.b.39b	R	$39$	$-1$	$3$	$3$	$0$	$-1$	$-1$	$-1$	$0$	$1$	$1$	$-1$	$-1$	$0$	$0$
11232.b.52a	R	$52$	$-4$	$0$	$-2$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed