Properties

Label 26T25
Order \(2704\)
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 $t$ :  $25$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5,7,9,11,13,2,4,6,8,10,12)(14,18,22,26,17,21,25,16,20,24,15,19,23), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7), (1,5,12,8)(2,10,11,3)(4,7,9,6)(14,21,25,18)(15,16,24,23)(17,19,22,20), (1,16,5,19)(2,20,4,15)(3,24)(6,23,13,25)(7,14,12,21)(8,18,11,17)(9,22,10,26)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$
16:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T25 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 $ $16$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $16$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $16$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
$ 13, 13 $ $16$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $8$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 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 $ $8$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $8$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$
$ 13, 13 $ $16$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $16$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $8$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $8$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 3)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(14,18,22,26,17,21,25,16,20,24,15, 19,23)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,22,17,25,20,15,23,18,26,21,16, 24,19)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $104$ $26$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,13)(11,12)(14,17,20,23,26,16,19,22,25,15,18, 21,24)$
$ 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 $ $338$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 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 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,16, 5,19)( 2,20, 4,15)( 3,24)( 6,23,13,25)( 7,14,12,21)( 8,18,11,17) ( 9,22,10,26)$
$ 26 $ $104$ $26$ $( 1,16,10,26, 6,23, 2,20,11,17, 7,14, 3,24,12,21, 8,18, 4,15,13,25, 9,22, 5,19 )$
$ 26 $ $104$ $26$ $( 1,20,13,16,12,25,11,21,10,17, 9,26, 8,22, 7,18, 6,14, 5,23, 4,19, 3,15, 2,24 )$
$ 26 $ $104$ $26$ $( 1,24, 3,19, 5,14, 7,22, 9,17,11,25,13,20, 2,15, 4,23, 6,18, 8,26,10,21,12,16 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,17)( 2,21)( 3,25)( 4,16)( 5,20)( 6,24)( 7,15)( 8,19)( 9,23)(10,14)(11,18) (12,22)(13,26)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,17,10,19)( 2,23, 9,26)( 3,16, 8,20)( 4,22, 7,14)( 5,15, 6,21)(11,25,13,24) (12,18)$
$ 26 $ $104$ $26$ $( 1,17, 5,15, 9,26,13,24, 4,22, 8,20,12,18, 3,16, 7,14,11,25, 2,23, 6,21,10,19 )$
$ 26 $ $104$ $26$ $( 1,21,12,22,10,23, 8,24, 6,25, 4,26, 2,14,13,15,11,16, 9,17, 7,18, 5,19, 3,20 )$
$ 26 $ $104$ $26$ $( 1,25, 6,16,11,20, 3,24, 8,15,13,19, 5,23,10,14, 2,18, 7,22,12,26, 4,17, 9,21 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,24)( 2,17)( 3,23)( 4,16)( 5,22)( 6,15)( 7,21)( 8,14)( 9,20)(10,26)(11,19) (12,25)(13,18)$

Group invariants

Order:  $2704=2^{4} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.