Properties

Label 26T29
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 $t$ :  $29$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,13)(4,12)(5,11)(6,10)(7,9)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21), (1,26,11,21,3,25,12,14,10,15,9,22,2,19,5,24,13,20,4,18,6,17,7,23)(8,16)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
8:  $C_8$
12:  $C_{12}$
24:  $C_{24}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T29 x 6

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

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.