Properties

Label 26T12
Order \(676\)
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 $t$ :  $12$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,2,23)(3,17,13,22)(4,24,12,15)(5,18,11,21)(6,25,10,14)(7,19,9,20)(8,26), (1,16,8,26)(2,23,7,19)(3,17,6,25)(4,24,5,18)(9,20,13,22)(10,14,12,15)(11,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
52:  $C_{13}:C_4$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T12 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 13, 13 $ $4$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $4$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $4$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $4$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $4$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $4$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 13 $ $4$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 4, 4, 4, 4, 4, 4, 2 $ $169$ $4$ $( 1,16, 2,23)( 3,17,13,22)( 4,24,12,15)( 5,18,11,21)( 6,25,10,14)( 7,19, 9,20) ( 8,26)$
$ 4, 4, 4, 4, 4, 4, 2 $ $169$ $4$ $( 1,25, 5,23)( 2,18, 4,17)( 3,24)( 6,16,13,19)( 7,22,12,26)( 8,15,11,20) ( 9,21,10,14)$

Group invariants

Order:  $676=2^{2} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [676, 11]
Character table: Data not available.