Properties

Label 26T20
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 $t$ :  $20$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,9,20,8,21,13,16)(2,14,4,25,7,22,5,24)(3,26,12,17,6,23,10,19)(11,18), (1,25,12,22,2,20,4,23)(3,15,9,24,13,17,7,21)(5,18,6,26,11,14,10,19)(8,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T20 x 6

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

Group invariants

Order:  $1352=2^{3} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1352, 40]
Character table: Data not available.