Properties

Label 26T47
Degree $26$
Order $11232$
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)
$|\Aut(F/K)|$:  $2$
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)

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

Group invariants

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