Group action invariants
| Degree $n$ : | $39$ | |
| Transitive number $t$ : | $14$ | |
| Group : | $D_{39}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| 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) | |
| $|\Aut(F/K)|$: | $1$ |
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
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