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

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.