Properties

Label 26T48
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 $t$:  $48$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
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)

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

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.