Properties

Label 26T34
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 $t$ :  $34$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,26,8,17,10,20,5,19,11,15,9,25)(2,21,12,23,13,18,4,24,7,22,6,14)(3,16), (1,6,9,3,2,4,13,8,5,11,12,10)(14,20,19,17,26,18,15,22,23,25,16,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $Q_8$
12:  $C_6\times C_2$
24:  24T4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T34 x 2

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 $ $24$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $24$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,16,18,20,22,24,26,15,17,19,21,23,25)$
$ 13, 13 $ $24$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $24$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $24$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $24$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 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 $ $338$ $12$ $( 1,26, 8,17,10,20, 5,19,11,15, 9,25)( 2,21,12,23,13,18, 4,24, 7,22, 6,14) ( 3,16)$
$ 12, 12, 2 $ $338$ $12$ $( 1,24, 6,14, 4,18,10,19, 5,16, 7,25)( 2,22, 3,20,13,26, 9,21, 8,23,11,17) (12,15)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,18, 2,25)( 3,19,13,24)( 4,26,12,17)( 5,20,11,23)( 6,14,10,16)( 7,21, 9,22) ( 8,15)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 12, 12, 2 $ $338$ $12$ $( 1,26)( 2,23,11,22,10,25,13,16, 4,17, 5,14)( 3,20, 8,18, 6,24,12,19, 7,21, 9, 15)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,24, 8,26)( 2,15, 7,22)( 3,19, 6,18)( 4,23, 5,14)( 9,17,13,20)(10,21,12,16) (11,25)$
$ 12, 12, 2 $ $338$ $12$ $( 1,18, 7,25, 5,14,10,22, 4,15, 6,26)( 2,17,11,21, 8,24, 9,23,13,19, 3,16) (12,20)$

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.