Properties

Label 26T41
Order \(6084\)
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 $t$ :  $41$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,7,25,2,26,4,23,11,19,3,18)(5,15,8,17,12,24,13,16,10,14,6,20)(9,22), (1,18,9,21,2,20,13,16,5,26,12,14)(3,22,4,24,8,19,11,25,10,23,6,15)(7,17)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$
36:  $C_3\times (C_3 : C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T45 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 $ $36$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $36$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,18,22,26,17,21,25,16,20,24,15,19,23)$
$ 13, 13 $ $36$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13 $ $36$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,22,17,25,20,15,23,18,26,21,16,24,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 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 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 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 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,26,17)(15,22,20)(16,18,23) (21,24,25)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,20)(15,21,23)(16,17,26) (18,22,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 9, 7)( 2,12, 3)( 4, 5, 8)( 6,11,13)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 3, 3, 3, 3, 1 $ $156$ $39$ $( 1, 4,13)( 2, 7, 9)( 3,10, 5)( 8,12,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,18,17,26,23,24)(16,22,20,25,19, 21)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 12, 12, 2 $ $507$ $12$ $( 1,21, 7,25, 2,26, 4,23,11,19, 3,18)( 5,15, 8,17,12,24,13,16,10,14, 6,20) ( 9,22)$
$ 12, 12, 2 $ $507$ $12$ $( 1,25,10,14, 7,22, 8,15,12,26, 2,18)( 3,24, 5,23,13,19, 6,16, 4,17, 9,21) (11,20)$
$ 4, 4, 4, 4, 4, 4, 2 $ $507$ $4$ $( 1,22,12,18)( 2,24,11,16)( 3,26,10,14)( 4,15, 9,25)( 5,17, 8,23)( 6,19, 7,21) (13,20)$
$ 4, 4, 4, 4, 4, 4, 2 $ $507$ $4$ $( 1,19, 8,18)( 2,17, 7,20)( 3,15, 6,22)( 4,26, 5,24)( 9,16,13,21)(10,14,12,23) (11,25)$
$ 12, 12, 2 $ $507$ $12$ $( 1,20,10,14, 9,19,12,17, 3,23, 4,18)( 2,15, 7,16, 5,26,11,22, 6,21, 8,24) (13,25)$
$ 12, 12, 2 $ $507$ $12$ $( 1,16,11,21,12,15, 3,17, 6,25, 5,18)( 2,23)( 4,24,10,14, 8,26,13,22, 7,19, 9, 20)$

Group invariants

Order:  $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.