Properties

Label 26T15
Order \(1014\)
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 $t$ :  $15$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,3,25,9,18)(2,24,6,15,5,14)(4,26,12,21,10,19)(7,16,8,17,11,20)(13,22), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
39:  $C_{13}:C_3$
78:  $C_{13}:C_6$, 26T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T15 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 $ $6$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $6$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $6$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $6$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $(14,25,23,21,19,17,15,26,24,22,20,18,16)$
$ 13, 13 $ $6$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $6$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $3$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $6$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $6$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $6$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $6$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $6$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $6$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 13, 13 $ $6$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $3$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $6$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $6$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $6$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $6$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $3$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $6$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $6$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $6$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $6$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $3$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $6$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,23, 3,25, 9,18)( 2,24, 6,15, 5,14)( 4,26,12,21,10,19)( 7,16, 8,17,11,20) (13,22)$
$ 26 $ $39$ $26$ $( 1,15,13,25,12,22,11,19,10,16, 9,26, 8,23, 7,20, 6,17, 5,14, 4,24, 3,21, 2,18 )$
$ 26 $ $39$ $26$ $( 1,19, 9,17, 4,15,12,26, 7,24, 2,22,10,20, 5,18,13,16, 8,14, 3,25,11,23, 6,21 )$
$ 26 $ $39$ $26$ $( 1,18,10,19, 6,20, 2,21,11,22, 7,23, 3,24,12,25, 8,26, 4,14,13,15, 9,16, 5,17 )$
$ 26 $ $39$ $26$ $( 1,21, 7,26,13,18, 6,23,12,15, 5,20,11,25, 4,17,10,22, 3,14, 9,19, 2,24, 8,16 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1,14)( 2,17)( 3,20)( 4,23)( 5,26)( 6,16)( 7,19)( 8,22)( 9,25)(10,15)(11,18) (12,21)(13,24)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,17,13,21, 4,18)( 2,26, 9,24, 7,19)( 3,22, 5,14,10,20)( 6,23) ( 8,15,11,16,12,25)$

Group invariants

Order:  $1014=2 \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1014, 11]
Character table: Data not available.