Properties

Label 26T18
Order \(1352\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
52:  $C_{13}:C_4$ x 2
104:  26T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T18 x 5

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

Group invariants

Order:  $1352=2^{3} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1352, 41]
Character table: Data not available.