LMFDB - Galois group: 26T48

Show commands: Magma

magma: G := TransitiveGroup(26, 48);

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 $t$:	$48$	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:	$(1,2)(3,23,19,22,8,14)(4,24,20,21,7,13)(5,6)(9,12,16,10,11,15)(17,25)(18,26)$, $(1,15,24,12,25,6,19,17,13,10,3,7,21,2,16,23,11,26,5,20,18,14,9,4,8,22)$	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: $C_2$
Degree 13: $\PSL(3,3)$

Low degree siblings

26T47 x 2, 26T48
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	$2^{8},1^{10}$	$117$	$2$	$8$	$( 1, 8)( 2, 7)( 9,21)(10,22)(11,18)(12,17)(23,26)(24,25)$
2C	$2^{13}$	$117$	$2$	$13$	$( 1, 7)( 2, 8)( 3, 4)( 5, 6)( 9,10)(11,26)(12,25)(13,15)(14,16)(17,24)(18,23)(19,20)(21,22)$
3A	$3^{6},1^{8}$	$104$	$3$	$12$	$( 5, 9,21)( 6,10,22)(11,13,18)(12,14,17)(15,23,26)(16,24,25)$
3B	$3^{8},1^{2}$	$624$	$3$	$16$	$( 1,25,13)( 2,26,14)( 5, 9,21)( 6,10,22)( 7,23,17)( 8,24,18)(11,19,16)(12,20,15)$
4A	$4^{4},2^{4},1^{2}$	$702$	$4$	$16$	$( 1,16, 8,13)( 2,15, 7,14)( 3, 5)( 4, 6)( 9,21)(10,22)(11,18,25,24)(12,17,26,23)$
4B	$4^{4},2^{5}$	$702$	$4$	$17$	$( 1,14, 8,15)( 2,13, 7,16)( 3, 6)( 4, 5)( 9,22)(10,21)(11,23,25,17)(12,24,26,18)(19,20)$
6A	$6^{3},2^{4}$	$104$	$6$	$19$	$( 1, 2)( 3, 4)( 5,22, 9, 6,21,10)( 7, 8)(11,17,13,12,18,14)(15,25,23,16,26,24)(19,20)$
6B	$6^{4},2$	$624$	$6$	$21$	$( 1,17,16, 2,18,15)( 3, 4)( 5,22, 9, 6,21,10)( 7,11,26, 8,12,25)(13,23,19,14,24,20)$
6C	$6^{3},2^{4}$	$936$	$6$	$19$	$( 1, 7)( 2, 8)( 3, 4)( 5,10,21, 6, 9,22)(11,15,18,26,13,23)(12,16,17,25,14,24)(19,20)$
6D	$6^{2},3^{2},2^{2},1^{4}$	$936$	$6$	$16$	$( 1,11,25, 8,18,24)( 2,12,26, 7,17,23)( 9,21)(10,22)(13,16,19)(14,15,20)$
8A1	$8^{2},4^{2},1^{2}$	$702$	$8$	$20$	$( 1,24,11,13, 8,18,25,16)( 2,23,12,14, 7,17,26,15)( 3, 9,21, 5)( 4,10,22, 6)$
8A-1	$8^{2},4^{2},1^{2}$	$702$	$8$	$20$	$( 1,25,24,13, 8,11,18,16)( 2,26,23,14, 7,12,17,15)( 3,21, 9, 5)( 4,22,10, 6)$
8B1	$8^{2},4^{2},2$	$702$	$8$	$21$	$( 1,12,24,15, 8,26,18,14)( 2,11,23,16, 7,25,17,13)( 3,22, 9, 6)( 4,21,10, 5)(19,20)$
8B-1	$8^{2},4^{2},2$	$702$	$8$	$21$	$( 1,17,11,15, 8,23,25,14)( 2,18,12,16, 7,24,26,13)( 3,10,21, 6)( 4, 9,22, 5)(19,20)$
13A1	$13^{2}$	$432$	$13$	$24$	$( 1, 8,11,13,21,16, 9,25,19, 5, 3,24,18)( 2, 7,12,14,22,15,10,26,20, 6, 4,23,17)$
13A-1	$13^{2}$	$432$	$13$	$24$	$( 1,25,24,19, 5, 3,18,16, 9,13,21,11, 8)( 2,26,23,20, 6, 4,17,15,10,14,22,12, 7)$
13A2	$13^{2}$	$432$	$13$	$24$	$( 1,18,13,21, 8,25,11,24,16, 9,19, 5, 3)( 2,17,14,22, 7,26,12,23,15,10,20, 6, 4)$
13A-2	$13^{2}$	$432$	$13$	$24$	$( 1,19, 5, 3,16, 9,24,11,18, 8,13,21,25)( 2,20, 6, 4,15,10,23,12,17, 7,14,22,26)$
26A1	$26$	$432$	$26$	$25$	$( 1, 7,24,15, 9,14,21,17,19, 6, 3,12,25, 2, 8,23,16,10,13,22,18,20, 5, 4,11,26)$
26A-1	$26$	$432$	$26$	$25$	$( 1,17,11,20, 5, 4,25,14,21,15, 9,23, 8, 2,18,12,19, 6, 3,26,13,22,16,10,24, 7)$
26A5	$26$	$432$	$26$	$25$	$( 1,20, 5, 4,13,22,11,23,25, 7,16,10,18, 2,19, 6, 3,14,21,12,24,26, 8,15, 9,17)$
26A-5	$26$	$432$	$26$	$25$	$( 1,26,16,10, 8,17,24,12,13,22,19, 6, 3, 2,25,15, 9, 7,18,23,11,14,21,20, 5, 4)$