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

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.