# Properties

 Label 26T16 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 $t$ : $16$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,19)(2,21,13,17)(3,23,12,15)(4,25,11,26)(5,14,10,24)(6,16,9,22)(7,18,8,20), (1,23,4,16,7,22,10,15,13,21,3,14,6,20,9,26,12,19,2,25,5,18,8,24,11,17) $|\Aut(F/K)|$: $1$

## 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, 26T19

Siblings are shown with degree $\leq 47$

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

## Conjugacy Classes

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

## Group invariants

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