Properties

Label 26T47
Order \(11232\)
n \(26\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $47$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,6,17,12,7,22,20)(2,13,5,18,11,8,21,19)(3,24,16,25,4,23,15,26)(9,10), (1,3,26,7,16,13,22,5)(2,4,25,8,15,14,21,6)(9,17,20,24,10,18,19,23)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5616:  $\PSL(3,3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 13: $\PSL(3,3)$

Low degree siblings

26T47, 26T48 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $117$ $2$ $( 1, 8)( 2, 7)( 3, 4)( 5,21)( 6,22)(11,14)(12,13)(15,26)(16,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $117$ $2$ $( 1, 7)( 2, 8)( 5,22)( 6,21)( 9,10)(11,13)(12,14)(15,25)(16,26)(17,18)(19,20) (23,24)$
$ 4, 4, 4, 4, 4, 4, 2 $ $702$ $4$ $( 1,14, 7,12)( 2,13, 8,11)( 3, 4)( 5,15,22,25)( 6,16,21,26)( 9,23,10,24) (17,20,18,19)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $702$ $4$ $( 1,13, 7,11)( 2,14, 8,12)( 5,16,22,26)( 6,15,21,25)( 9,24,10,23)(17,19,18,20)$
$ 8, 8, 8, 2 $ $702$ $8$ $( 1,25,13, 6, 7,15,11,21)( 2,26,14, 5, 8,16,12,22)( 3, 4)( 9,20,24,17,10,19, 23,18)$
$ 8, 8, 8, 1, 1 $ $702$ $8$ $( 1,26,13, 5, 7,16,11,22)( 2,25,14, 6, 8,15,12,21)( 9,19,24,18,10,20,23,17)$
$ 8, 8, 8, 2 $ $702$ $8$ $( 1,21,11,15, 7, 6,13,25)( 2,22,12,16, 8, 5,14,26)( 3, 4)( 9,18,23,19,10,17, 24,20)$
$ 8, 8, 8, 1, 1 $ $702$ $8$ $( 1,22,11,16, 7, 5,13,26)( 2,21,12,15, 8, 6,14,25)( 9,17,23,20,10,18,24,19)$
$ 26 $ $432$ $26$ $( 1, 7,21,10,19,16, 3,17, 5,24,14,26,12, 2, 8,22, 9,20,15, 4,18, 6,23,13,25,11 )$
$ 13, 13 $ $432$ $13$ $( 1, 8,21, 9,19,15, 3,18, 5,23,14,25,12)( 2, 7,22,10,20,16, 4,17, 6,24,13,26, 11)$
$ 26 $ $432$ $26$ $( 1, 6, 9,26, 3, 7,23,20,12,17,21,13,15, 2, 5,10,25, 4, 8,24,19,11,18,22,14,16 )$
$ 13, 13 $ $432$ $13$ $( 1, 5, 9,25, 3, 8,23,19,12,18,21,14,15)( 2, 6,10,26, 4, 7,24,20,11,17,22,13, 16)$
$ 13, 13 $ $432$ $13$ $( 1,12,25,14,23, 5,18, 3,15,19, 9,21, 8)( 2,11,26,13,24, 6,17, 4,16,20,10,22, 7)$
$ 26 $ $432$ $26$ $( 1,11,25,13,23, 6,18, 4,15,20, 9,22, 8, 2,12,26,14,24, 5,17, 3,16,19,10,21, 7 )$
$ 26 $ $432$ $26$ $( 1,16,14,22,18,11,19,24, 8, 4,25,10, 5, 2,15,13,21,17,12,20,23, 7, 3,26, 9, 6 )$
$ 13, 13 $ $432$ $13$ $( 1,15,14,21,18,12,19,23, 8, 3,25, 9, 5)( 2,16,13,22,17,11,20,24, 7, 4,26,10, 6)$
$ 6, 6, 6, 2, 2, 2, 2 $ $104$ $6$ $( 1, 2)( 3, 4)( 5, 9,21, 6,10,22)( 7, 8)(11,13,17,12,14,18)(15,23,26,16,24,25) (19,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $104$ $3$ $( 5,10,21)( 6, 9,22)(11,14,17)(12,13,18)(15,24,26)(16,23,25)$
$ 6, 6, 6, 2, 2, 2, 1, 1 $ $936$ $6$ $( 1, 8)( 2, 7)( 3, 4)( 5,26,10,15,21,24)( 6,25, 9,16,22,23)(11,18,14,12,17,13)$
$ 6, 6, 3, 3, 2, 2, 2, 1, 1 $ $936$ $6$ $( 1, 7)( 2, 8)( 5,25,10,16,21,23)( 6,26, 9,15,22,24)(11,17,14)(12,18,13) (19,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $624$ $3$ $( 1, 6,23)( 2, 5,24)( 3,15,10)( 4,16, 9)( 7,14,18)( 8,13,17)(11,22,26) (12,21,25)$
$ 6, 6, 6, 6, 2 $ $624$ $6$ $( 1, 5,23, 2, 6,24)( 3,16,10, 4,15, 9)( 7,13,18, 8,14,17)(11,21,26,12,22,25) (19,20)$

Group invariants

Order:  $11232=2^{5} \cdot 3^{3} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.