Properties

Label 26T22
Order \(2028\)
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 $t$ :  $22$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,8,23,2,14,9,18,3,22,10,26,4,17,11,21,5,25,12,16,6,20,13,24,7,15), (1,5)(2,4)(6,13)(7,12)(8,11)(9,10)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21), (1,8,10,5,11,9)(2,12,13,4,7,6)(14,26,22,19,20,24)(15,17,25,18,16,21)
$|\Aut(F/K)|$:  $1$

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
12:  $C_6\times C_2$
78:  $C_{13}:C_6$ x 2
156:  26T9 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T22 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 $ $6$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $6$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,24,21,18,15,25,22,19,16,26,23,20,17)$
$ 13, 13 $ $12$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $6$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 13 $ $12$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $12$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $12$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $12$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,21,15,22,16,23,17,24,18,25,19,26,20)$
$ 13, 13 $ $12$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 13 $ $12$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $6$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$
$ 13, 13 $ $12$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 13, 13 $ $12$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $12$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 26 $ $78$ $26$ $( 1,19, 8,23, 2,14, 9,18, 3,22,10,26, 4,17,11,21, 5,25,12,16, 6,20,13,24, 7,15 )$
$ 26 $ $78$ $26$ $( 1,14,10,21, 6,15, 2,22,11,16, 7,23, 3,17,12,24, 8,18, 4,25,13,19, 9,26, 5,20 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1,17)( 2,25)( 3,20)( 4,15)( 5,23)( 6,18)( 7,26)( 8,21)( 9,16)(10,24)(11,19) (12,14)(13,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 26 $ $78$ $26$ $( 1,19,10,25, 6,18, 2,24,11,17, 7,23, 3,16,12,22, 8,15, 4,21,13,14, 9,20, 5,26 )$
$ 26 $ $78$ $26$ $( 1,16,12,19,10,22, 8,25, 6,15, 4,18, 2,21,13,24,11,14, 9,17, 7,20, 5,23, 3,26 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1,26)( 2,18)( 3,23)( 4,15)( 5,20)( 6,25)( 7,17)( 8,22)( 9,14)(10,19)(11,24) (12,16)(13,21)$
$ 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, 2 $ $169$ $6$ $( 1,19, 6,22,12,23)( 2,17)( 3,15,11,25, 5,24)( 4,26, 7,20, 8,18) ( 9,16,13,21,10,14)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,19,12,15, 7,18)( 2,21, 8,20,10,24)( 3,23, 4,25,13,17)( 5,14, 9,22, 6,16) (11,26)$
$ 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, 2 $ $169$ $6$ $( 1,19,13,25,10,17)( 2,26, 3,20, 6,15)( 4,14, 9,23,11,24)( 5,21,12,18, 7,22) ( 8,16)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 6, 6, 6, 6, 2 $ $169$ $6$ $( 1,19, 5,17, 4,24)( 2,25, 8,22,13,26)( 3,18,11,14, 9,15)( 6,23, 7,16,10,21) (12,20)$

Group invariants

Order:  $2028=2^{2} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.