↑

Properties

Label 26T19
Order \(1352\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $19$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,3,15,5,24,7,20,9,16,11,25,13,21,2,17,4,26,6,22,8,18,10,14,12,23), (1,12,2,4)(3,9,13,7)(5,6,11,10)(14,23,17,21)(15,18,16,26)(19,24,25,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T16 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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 $ $1$ $1$ $()$
$ 13, 13 $ $4$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $4$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,15,16,17,18,19,20,21,22,23,24,25,26)$
$ 13, 13 $ $8$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $8$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 13 $ $8$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $8$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,16,18,20,22,24,26,15,17,19,21,23,25)$
$ 13, 13 $ $8$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 13 $ $8$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $8$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,18,22,26,17,21,25,16,20,24,15,19,23)$
$ 13, 13 $ $8$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $8$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $8$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $8$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $8$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $4$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $4$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 26 $ $52$ $26$ $( 1,19, 3,15, 5,24, 7,20, 9,16,11,25,13,21, 2,17, 4,26, 6,22, 8,18,10,14,12,23 )$
$ 26 $ $52$ $26$ $( 1,15, 7,16,13,17, 6,18,12,19, 5,20,11,21, 4,22,10,23, 3,24, 9,25, 2,26, 8,14 )$
$ 26 $ $52$ $26$ $( 1,24,11,17, 8,23, 5,16, 2,22,12,15, 9,21, 6,14, 3,20,13,26,10,19, 7,25, 4,18 )$
$ 26 $ $52$ $26$ $( 1,16, 6,19,11,22, 3,25, 8,15,13,18, 5,21,10,24, 2,14, 7,17,12,20, 4,23, 9,26 )$
$ 26 $ $52$ $26$ $( 1,20, 2,18, 3,16, 4,14, 5,25, 6,23, 7,21, 8,19, 9,17,10,15,11,26,12,24,13,22 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,21)( 2,19)( 3,17)( 4,15)( 5,26)( 6,24)( 7,22)( 8,20)( 9,18)(10,16)(11,14) (12,25)(13,23)$
$ 26 $ $52$ $26$ $( 1,25,10,20, 6,15, 2,23,11,18, 7,26, 3,21,12,16, 8,24, 4,19,13,14, 9,22, 5,17 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 26 $ $52$ $26$ $( 1,19, 5,18, 9,17,13,16, 4,15, 8,14,12,26, 3,25, 7,24,11,23, 2,22, 6,21,10,20 )$
$ 26 $ $52$ $26$ $( 1,24, 2,14, 3,17, 4,20, 5,23, 6,26, 7,16, 8,19, 9,22,10,25,11,15,12,18,13,21 )$
$ 26 $ $52$ $26$ $( 1,16,12,23,10,17, 8,24, 6,18, 4,25, 2,19,13,26,11,20, 9,14, 7,21, 5,15, 3,22 )$
$ 26 $ $52$ $26$ $( 1,26, 6,15,11,17, 3,19, 8,21,13,23, 5,25,10,14, 2,16, 7,18,12,20, 4,22, 9,24 )$
$ 26 $ $52$ $26$ $( 1,20, 7,25,13,17, 6,22,12,14, 5,19,11,24, 4,16,10,21, 3,26, 9,18, 2,23, 8,15 )$
$ 26 $ $52$ $26$ $( 1,22,11,26, 8,17, 5,21, 2,25,12,16, 9,20, 6,24, 3,15,13,19,10,23, 7,14, 4,18 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,17)( 2,20)( 3,23)( 4,26)( 5,16)( 6,19)( 7,22)( 8,25)( 9,15)(10,18)(11,21) (12,24)(13,14)$

Group invariants

Order:  $1352=2^{3} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1352, 43]
Character table: Data not available.