Properties

Label 26T48
Order \(11232\)
n \(26\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $48$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,23,19,22,8,14)(4,24,20,21,7,13)(5,6)(9,12,16,10,11,15)(17,25)(18,26), (1,15,24,12,25,6,19,17,13,10,3,7,21,2,16,23,11,26,5,20,18,14,9,4,8,22)
$|\Aut(F/K)|$:  $2$

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: $C_2$

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

Low degree siblings

26T47 x 2, 26T48

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

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.