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

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.