Properties

 Label 39T29 Order $$1053$$ n $$39$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No

Group action invariants

 Degree $n$ : $39$ Transitive number $t$ : $29$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,15,26,38,10,23,34,9,20,31,6,18,30)(2,13,27,39,11,24,35,7,21,32,4,16,28)(3,14,25,37,12,22,36,8,19,33,5,17,29), (1,3,2)(4,10,28,5,11,29,6,12,30)(7,20,18,8,21,16,9,19,17)(13,38,31,15,37,33,14,39,32)(22,25,35)(23,26,36)(24,27,34) $|\Aut(F/K)|$: $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
39:  $C_{13}:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $C_{13}:C_3$

Low degree siblings

27T292, 39T30 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $13$ $3$ $( 4, 6, 5)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,26,27) (34,35,36)(37,39,38)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $13$ $3$ $( 4, 5, 6)(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26) (34,36,35)(37,38,39)$ $13, 13, 13$ $81$ $13$ $( 1,15,26,38,10,23,34, 9,20,31, 6,18,30)( 2,13,27,39,11,24,35, 7,21,32, 4,16, 28)( 3,14,25,37,12,22,36, 8,19,33, 5,17,29)$ $13, 13, 13$ $81$ $13$ $( 1,26,10,34,20, 6,30,15,38,23, 9,31,18)( 2,27,11,35,21, 4,28,13,39,24, 7,32, 16)( 3,25,12,36,19, 5,29,14,37,22, 8,33,17)$ $13, 13, 13$ $81$ $13$ $( 1,10,20,30,38, 9,18,26,34, 6,15,23,31)( 2,11,21,28,39, 7,16,27,35, 4,13,24, 32)( 3,12,19,29,37, 8,17,25,36, 5,14,22,33)$ $13, 13, 13$ $81$ $13$ $( 1,20,38,18,34,15,31,10,30, 9,26, 6,23)( 2,21,39,16,35,13,32,11,28, 7,27, 4, 24)( 3,19,37,17,36,14,33,12,29, 8,25, 5,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $117$ $3$ $( 1,35,16)( 2,36,17)( 3,34,18)( 7,14,32)( 8,15,33)( 9,13,31)(10,23,21) (11,24,19)(12,22,20)(25,28,38)(26,29,39)(27,30,37)$ $9, 9, 9, 3, 3, 3, 3$ $117$ $9$ $( 1,36,18, 3,35,17, 2,34,16)( 4, 6, 5)( 7,15,33, 8,13,31, 9,14,32) (10,24,21,11,22,19,12,23,20)(25,28,37)(26,29,38)(27,30,39)$ $9, 9, 9, 3, 3, 3, 3$ $117$ $9$ $( 1,34,17, 2,35,18, 3,36,16)( 4, 5, 6)( 7,13,31, 9,15,33, 8,14,32) (10,22,21,12,24,20,11,23,19)(25,28,39)(26,29,37)(27,30,38)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $117$ $3$ $( 1,16,35)( 2,17,36)( 3,18,34)( 7,32,14)( 8,33,15)( 9,31,13)(10,21,23) (11,19,24)(12,20,22)(25,38,28)(26,39,29)(27,37,30)$ $9, 9, 9, 3, 3, 3, 3$ $117$ $9$ $( 1,17,34, 3,16,36, 2,18,35)( 4, 6, 5)( 7,32,15, 8,33,13, 9,31,14) (10,20,23,11,21,24,12,19,22)(25,37,30)(26,38,28)(27,39,29)$ $9, 9, 9, 3, 3, 3, 3$ $117$ $9$ $( 1,18,36, 2,16,34, 3,17,35)( 4, 5, 6)( 7,32,13, 9,31,15, 8,33,14) (10,19,23,12,21,22,11,20,24)(25,39,29)(26,37,30)(27,38,28)$

Group invariants

 Order: $1053=3^{4} \cdot 13$ Cyclic: No Abelian: No Solvable: Yes GAP id: [1053, 51]
 Character table:  3 4 4 4 . . . . 2 2 2 2 2 2 13 1 . . 1 1 1 1 . . . . . . 1a 3a 3b 13a 13b 13c 13d 3c 9a 9b 3d 9c 9d 2P 1a 3b 3a 13b 13c 13d 13a 3d 9d 9c 3c 9b 9a 3P 1a 1a 1a 13a 13b 13c 13d 1a 3a 3b 1a 3a 3b 5P 1a 3b 3a 13b 13c 13d 13a 3d 9d 9c 3c 9b 9a 7P 1a 3a 3b 13d 13a 13b 13c 3c 9a 9b 3d 9c 9d 11P 1a 3b 3a 13d 13a 13b 13c 3d 9d 9c 3c 9b 9a 13P 1a 3a 3b 1a 1a 1a 1a 3c 9a 9b 3d 9c 9d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 1 1 D D D /D /D /D X.3 1 1 1 1 1 1 1 /D /D /D D D D X.4 3 3 3 B C /B /C . . . . . . X.5 3 3 3 /B /C B C . . . . . . X.6 3 3 3 C /B /C B . . . . . . X.7 3 3 3 /C B C /B . . . . . . X.8 13 A /A . . . . D /D 1 /D 1 D X.9 13 /A A . . . . /D D 1 D 1 /D X.10 13 A /A . . . . /D 1 D D /D 1 X.11 13 /A A . . . . D 1 /D /D D 1 X.12 13 A /A . . . . 1 D /D 1 D /D X.13 13 /A A . . . . 1 /D D 1 /D D A = 2*E(3)-E(3)^2 = (-1+3*Sqrt(-3))/2 = 1+3b3 B = E(13)^2+E(13)^5+E(13)^6 C = E(13)^4+E(13)^10+E(13)^12 D = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3