Properties

Label 26T48
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 $t$ :  $48$
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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $117$ $2$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,24)( 6,23)(19,21)(20,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $117$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,23)( 6,24)( 7, 8)( 9,10)(11,12)(13,14)(19,22) (20,21)(25,26)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ $702$ $4$ $( 1, 3,16,18)( 2, 4,15,17)( 5,19,24,21)( 6,20,23,22)( 7,26)( 8,25)( 9,13) (10,14)$
$ 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $702$ $4$ $( 1, 4,16,17)( 2, 3,15,18)( 5,20,24,22)( 6,19,23,21)( 7,25)( 8,26)( 9,14) (10,13)(11,12)$
$ 8, 8, 4, 4, 1, 1 $ $702$ $8$ $( 1, 5, 3,19,16,24,18,21)( 2, 6, 4,20,15,23,17,22)( 7,14,26,10)( 8,13,25, 9)$
$ 8, 8, 4, 4, 2 $ $702$ $8$ $( 1, 6, 3,20,16,23,18,22)( 2, 5, 4,19,15,24,17,21)( 7,13,26, 9)( 8,14,25,10) (11,12)$
$ 8, 8, 4, 4, 1, 1 $ $702$ $8$ $( 1,21,18,24,16,19, 3, 5)( 2,22,17,23,15,20, 4, 6)( 7,10,26,14)( 8, 9,25,13)$
$ 8, 8, 4, 4, 2 $ $702$ $8$ $( 1,22,18,23,16,20, 3, 6)( 2,21,17,24,15,19, 4, 5)( 7, 9,26,13)( 8,10,25,14) (11,12)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $624$ $3$ $( 1,21,25)( 2,22,26)( 3, 8,16)( 4, 7,15)( 5,13,11)( 6,14,12)(17,23,20) (18,24,19)$
$ 6, 6, 6, 6, 2 $ $624$ $6$ $( 1,22,25, 2,21,26)( 3, 7,16, 4, 8,15)( 5,14,11, 6,13,12)( 9,10) (17,24,20,18,23,19)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $104$ $3$ $( 1,11,16)( 2,12,15)( 3,21, 5)( 4,22, 6)( 7,26,14)( 8,25,13)$
$ 6, 6, 6, 2, 2, 2, 2 $ $104$ $6$ $( 1,12,16, 2,11,15)( 3,22, 5, 4,21, 6)( 7,25,14, 8,26,13)( 9,10)(17,18)(19,20) (23,24)$
$ 6, 6, 3, 3, 2, 2, 1, 1, 1, 1 $ $936$ $6$ $( 1,16)( 2,15)( 3,24,21,18, 5,19)( 4,23,22,17, 6,20)( 7,26,14)( 8,25,13)$
$ 6, 6, 6, 2, 2, 2, 2 $ $936$ $6$ $( 1,15)( 2,16)( 3,23,21,17, 5,20)( 4,24,22,18, 6,19)( 7,25,14, 8,26,13)( 9,10) (11,12)$
$ 13, 13 $ $432$ $13$ $( 1, 9,19, 3,21,11,18, 8, 5,25,13,16,24)( 2,10,20, 4,22,12,17, 7, 6,26,14,15, 23)$
$ 26 $ $432$ $26$ $( 1,10,19, 4,21,12,18, 7, 5,26,13,15,24, 2, 9,20, 3,22,11,17, 8, 6,25,14,16,23 )$
$ 13, 13 $ $432$ $13$ $( 1, 5, 3,16,18, 9,25,21,24, 8,19,13,11)( 2, 6, 4,15,17,10,26,22,23, 7,20,14, 12)$
$ 26 $ $432$ $26$ $( 1, 6, 3,15,18,10,25,22,24, 7,19,14,11, 2, 5, 4,16,17, 9,26,21,23, 8,20,13,12 )$
$ 13, 13 $ $432$ $13$ $( 1,24,16,13,25, 5, 8,18,11,21, 3,19, 9)( 2,23,15,14,26, 6, 7,17,12,22, 4,20, 10)$
$ 26 $ $432$ $26$ $( 1,23,16,14,25, 6, 8,17,11,22, 3,20, 9, 2,24,15,13,26, 5, 7,18,12,21, 4,19,10 )$
$ 13, 13 $ $432$ $13$ $( 1,11,13,19, 8,24,21,25, 9,18,16, 3, 5)( 2,12,14,20, 7,23,22,26,10,17,15, 4, 6)$
$ 26 $ $432$ $26$ $( 1,12,13,20, 8,23,21,26, 9,17,16, 4, 5, 2,11,14,19, 7,24,22,25,10,18,15, 3, 6 )$

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.