# Properties

 Label 39T14 Degree $39$ Order $234$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{39}:C_3$

## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $14$ Group: $D_{39}:C_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,34,12,33,38,24)(2,36,10,32,39,23)(3,35,11,31,37,22)(4,7,19,30,26,13)(5,9,20,29,27,15)(6,8,21,28,25,14)(17,18), (1,29,39)(2,30,37)(3,28,38)(4,16,7)(5,17,8)(6,18,9)(10,31,26)(11,32,27)(12,33,25)(13,19,36)(14,20,34)(15,21,35)(22,24,23)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$78$:  $C_{13}:C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_6$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $234=2 \cdot 3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [234, 9]
 Character table:  2 1 1 1 1 1 1 . . . . . . . . . 3 2 2 2 1 1 1 2 2 2 1 1 1 1 1 1 13 1 . . . . . 1 . . 1 1 1 1 1 1 1a 3a 3b 6a 6b 2a 3c 3d 3e 39a 13a 39b 13b 39c 39d 2P 1a 3b 3a 3a 3b 1a 3c 3e 3d 39c 13b 39d 13a 39b 39a 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 13a 13a 13a 13b 13b 13b 5P 1a 3b 3a 6b 6a 2a 3c 3e 3d 39c 13b 39d 13a 39b 39a 7P 1a 3a 3b 6a 6b 2a 3c 3d 3e 39c 13b 39d 13a 39b 39a 11P 1a 3b 3a 6b 6a 2a 3c 3e 3d 39d 13b 39c 13a 39a 39b 13P 1a 3a 3b 6a 6b 2a 3c 3d 3e 3c 1a 3c 1a 3c 3c 17P 1a 3b 3a 6b 6a 2a 3c 3e 3d 39a 13a 39b 13b 39c 39d 19P 1a 3a 3b 6a 6b 2a 3c 3d 3e 39d 13b 39c 13a 39a 39b 23P 1a 3b 3a 6b 6a 2a 3c 3e 3d 39a 13a 39b 13b 39c 39d 29P 1a 3b 3a 6b 6a 2a 3c 3e 3d 39b 13a 39a 13b 39d 39c 31P 1a 3a 3b 6a 6b 2a 3c 3d 3e 39d 13b 39c 13a 39a 39b 37P 1a 3a 3b 6a 6b 2a 3c 3d 3e 39c 13b 39d 13a 39b 39a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 X.3 1 A /A -/A -A -1 1 A /A 1 1 1 1 1 1 X.4 1 /A A -A -/A -1 1 /A A 1 1 1 1 1 1 X.5 1 A /A /A A 1 1 A /A 1 1 1 1 1 1 X.6 1 /A A A /A 1 1 /A A 1 1 1 1 1 1 X.7 2 2 2 . . . -1 -1 -1 -1 2 -1 2 -1 -1 X.8 2 B /B . . . -1 -/A -A -1 2 -1 2 -1 -1 X.9 2 /B B . . . -1 -A -/A -1 2 -1 2 -1 -1 X.10 6 . . . . . 6 . . C C C *C *C *C X.11 6 . . . . . 6 . . *C *C *C C C C X.12 6 . . . . . -3 . . D C F *C G E X.13 6 . . . . . -3 . . E *C G C D F X.14 6 . . . . . -3 . . F C D *C E G X.15 6 . . . . . -3 . . G *C E C F D A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3 C = E(13)^2+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^11 = (-1-Sqrt(13))/2 = -1-b13 D = E(39)^8+E(39)^11+E(39)^19+E(39)^20+E(39)^28+E(39)^31 E = E(39)^4+E(39)^10+E(39)^14+E(39)^25+E(39)^29+E(39)^35 F = E(39)^2+E(39)^5+E(39)^7+E(39)^32+E(39)^34+E(39)^37 G = E(39)+E(39)^16+E(39)^17+E(39)^22+E(39)^23+E(39)^38