Properties

Label 26T33
Order \(4056\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $33$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,12,25,4,20,11,26,13,24,8,16)(2,22,3,21,7,17,10,14,9,15,5,19)(6,18), (1,16,2,21,11,14)(3,26,7,20,4,18)(5,23,12,19,10,22)(6,15,8,25,13,24)(9,17)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_4$ x 2, $C_2^2$
6:  $C_6$ x 3
8:  $C_4\times C_2$
12:  $C_{12}$ x 2, $C_6\times C_2$
24:  24T2
156:  $F_{13}$ x 2
312:  26T10 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T33 x 5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

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 $ $1$ $1$ $()$
$ 13, 13 $ $24$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,24,21,18,15,25,22,19,16,26,23,20,17)$
$ 13, 13 $ $24$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $24$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $24$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $24$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $12$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $12$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 12, 12, 2 $ $169$ $12$ $( 1,23,12,25, 4,20,11,26,13,24, 8,16)( 2,22, 3,21, 7,17,10,14, 9,15, 5,19) ( 6,18)$
$ 12, 12, 2 $ $169$ $12$ $( 1,15,13,18, 3,22, 7,23, 8,20, 5,16)( 2,25,10,14,12,21, 6,26,11,24, 9,17) ( 4,19)$
$ 4, 4, 4, 4, 4, 4, 2 $ $169$ $4$ $( 1,17, 4,16)( 2,21, 3,25)( 5,20,13,26)( 6,24,12,22)( 7,15,11,18)( 8,19,10,14) ( 9,23)$
$ 4, 4, 4, 4, 4, 4, 2 $ $169$ $4$ $( 1,24, 3,16)( 2,20)( 4,25,13,15)( 5,21,12,19)( 6,17,11,23)( 7,26,10,14) ( 8,22, 9,18)$
$ 12, 12, 2 $ $169$ $12$ $( 1,18, 9,26, 2,19,13,17, 5,22,12,16)( 3,20, 4,21, 8,25,11,15,10,14, 6,23) ( 7,24)$
$ 12, 12, 2 $ $169$ $12$ $( 1,26, 4,22, 8,21, 9,24, 6,15, 2,16)( 3,19,11,17,13,23, 7,18,12,20,10,14) ( 5,25)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,25,18,19,17,21,26,16,23,22,24,20)$
$ 12, 12, 1, 1 $ $169$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1,23)( 2,17)( 3,24)( 4,18)( 5,25)( 6,19)( 7,26)( 8,20)( 9,14)(10,21)(11,15) (12,22)(13,16)$
$ 26 $ $156$ $26$ $( 1,22,11,14, 8,19, 5,24, 2,16,12,21, 9,26, 6,18, 3,23,13,15,10,20, 7,25, 4,17 )$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,15, 5,21, 2,23)( 3,18,10,22, 8,19)( 4,26, 6,16,11,17)( 7,24) ( 9,14,12,25,13,20)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,17,10,25,11,23)( 2,15,13,19, 7,18)( 3,26)( 4,24, 6,20,12,21) ( 5,22, 9,14, 8,16)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,24, 9,14, 7,23)( 2,26,12,20, 3,15)( 4,17, 5,19, 8,25)( 6,21,11,18,13,22) (10,16)$
$ 26 $ $156$ $26$ $( 1,18,11,26, 8,21, 5,16, 2,24,12,19, 9,14, 6,22, 3,17,13,25,10,20, 7,15, 4,23 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1,15)( 2,21)( 3,14)( 4,20)( 5,26)( 6,19)( 7,25)( 8,18)( 9,24)(10,17)(11,23) (12,16)(13,22)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,26, 9,14, 3,23)( 2,18, 5,20, 6,25)( 4,15,10,19,12,16)( 7,17,11,24, 8,22) (13,21)$

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.