LMFDB - Galois group: 26T47

Show commands: Magma

magma: G := TransitiveGroup(26, 47);

Group invariants

Abstract group:	$\GL(3,3)$	magma: IdentifyGroup(G);
Order:	$11232=2^{5} \cdot 3^{3} \cdot 13$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	no	magma: IsSolvable(G);
Nilpotency class:	not nilpotent	magma: NilpotencyClass(G);

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

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

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	Index	Representative
1A	$1^{26}$	$1$	$1$	$0$	$()$
2A	$2^{13}$	$1$	$2$	$13$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)$
2B	$2^{12},1^{2}$	$117$	$2$	$12$	$( 1, 9)( 2,10)( 3, 4)( 5, 7)( 6, 8)(13,18)(14,17)(15,16)(19,22)(20,21)(23,24)(25,26)$
2C	$2^{9},1^{8}$	$117$	$2$	$9$	$( 1, 9)( 2,10)( 5,20)( 6,19)( 7,21)( 8,22)(15,16)(23,26)(24,25)$
3A	$3^{6},1^{8}$	$104$	$3$	$12$	$( 1, 8,19)( 2, 7,20)( 5,10,21)( 6, 9,22)(11,17,14)(12,18,13)$
3B	$3^{8},1^{2}$	$624$	$3$	$16$	$( 1,18,19)( 2,17,20)( 3,25,24)( 4,26,23)( 5,10, 7)( 6, 9, 8)(11,21,14)(12,22,13)$
4A	$4^{6},1^{2}$	$702$	$4$	$18$	$( 1,24,10,26)( 2,23, 9,25)( 3,11, 4,12)( 5,21,19, 8)( 6,22,20, 7)(13,18,14,17)$
4B	$4^{6},2$	$702$	$4$	$19$	$( 1,25,10,23)( 2,26, 9,24)( 3,11, 4,12)( 5, 7,19,22)( 6, 8,20,21)(13,18,14,17)(15,16)$
6A	$6^{3},2^{4}$	$104$	$6$	$19$	$( 1,20, 8, 2,19, 7)( 3, 4)( 5,22,10, 6,21, 9)(11,13,17,12,14,18)(15,16)(23,24)(25,26)$
6B	$6^{4},2$	$624$	$6$	$21$	$( 1,14, 8, 2,13, 7)( 3,23,25, 4,24,26)( 5,18,11, 6,17,12)( 9,20,22,10,19,21)(15,16)$
6C	$6^{2},3^{2},2^{3},1^{2}$	$936$	$6$	$17$	$( 1,22,19, 9, 8, 6)( 2,21,20,10, 7, 5)(11,17,14)(12,18,13)(15,16)(23,26)(24,25)$
6D	$6^{3},2^{3},1^{2}$	$936$	$6$	$18$	$( 1, 9)( 2,10)( 3,23,25, 4,24,26)( 5,21,14, 7,20,17)( 6,22,13, 8,19,18)(15,16)$
8A1	$8^{3},2$	$702$	$8$	$22$	$( 1,23, 8,20,10,25,21, 6)( 2,24, 7,19, 9,26,22, 5)( 3,17,13,12, 4,18,14,11)(15,16)$
8A-1	$8^{3},2$	$702$	$8$	$22$	$( 1,23,19,22,10,25, 5, 7)( 2,24,20,21, 9,26, 6, 8)( 3,14,17,12, 4,13,18,11)(15,16)$
8B1	$8^{3},1^{2}$	$702$	$8$	$21$	$( 1,26,19, 8,10,24, 5,21)( 2,25,20, 7, 9,23, 6,22)( 3,14,17,12, 4,13,18,11)$
8B-1	$8^{3},1^{2}$	$702$	$8$	$21$	$( 1,26, 8, 5,10,24,21,19)( 2,25, 7, 6, 9,23,22,20)( 3,17,13,12, 4,18,14,11)$
13A1	$13^{2}$	$432$	$13$	$24$	$( 1,24, 5,20,21, 8,14,11,15,25,17, 9, 4)( 2,23, 6,19,22, 7,13,12,16,26,18,10, 3)$
13A-1	$13^{2}$	$432$	$13$	$24$	$( 1, 3,10,25,17,12,16,23,19,22, 7, 6,13)( 2, 4, 9,26,18,11,15,24,20,21, 8, 5,14)$
13A2	$13^{2}$	$432$	$13$	$24$	$( 1,25,21,18,11,15, 3,10,24, 5,20, 7,13)( 2,26,22,17,12,16, 4, 9,23, 6,19, 8,14)$
13A-2	$13^{2}$	$432$	$13$	$24$	$( 1,24,20, 7, 6,13, 9, 4,12,16,26,22,17)( 2,23,19, 8, 5,14,10, 3,11,15,25,21,18)$
26A1	$26$	$432$	$26$	$25$	$( 1,25,21, 8, 5,20,17,12,15,24,14,10, 3, 2,26,22, 7, 6,19,18,11,16,23,13, 9, 4)$
26A-1	$26$	$432$	$26$	$25$	$( 1, 3,10,24,14,11,16,26, 8, 5,20,21,18, 2, 4, 9,23,13,12,15,25, 7, 6,19,22,17)$
26A5	$26$	$432$	$26$	$25$	$( 1,25, 7,19,22,17, 9, 4,12,15,24, 5,14, 2,26, 8,20,21,18,10, 3,11,16,23, 6,13)$
26A-5	$26$	$432$	$26$	$25$	$( 1,24, 5,14,11,16, 4, 9,26,22, 7,19,18, 2,23, 6,13,12,15, 3,10,25,21, 8,20,17)$