Group action invariants
| Degree $n$: | $39$ | |
| Transitive number $t$: | $9$ | |
| Group: | $D_{13}.S_3$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,14,26,37,12,22,36,9,19,31,6,16,29,2,15,27,38,10,23,34,7,20,32,4,17,30,3,13,25,39,11,24,35,8,21,33,5,18,28), (1,20,31,13)(2,19,32,15)(3,21,33,14)(4,34,28,38)(5,36,29,37)(6,35,30,39)(7,12,26,24)(8,11,27,23)(9,10,25,22)(16,18) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $6$: $S_3$ $12$: $C_3 : C_4$ $52$: $C_{13}:C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 13: $C_{13}:C_4$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $13$ | $2$ | $( 4,39)( 5,37)( 6,38)( 7,35)( 8,36)( 9,34)(10,33)(11,31)(12,32)(13,29)(14,30) (15,28)(16,26)(17,27)(18,25)(19,22)(20,23)(21,24)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $39$ | $4$ | $( 2, 3)( 4,18,39,25)( 5,17,37,27)( 6,16,38,26)( 7,32,35,12)( 8,31,36,11) ( 9,33,34,10)(13,23,29,20)(14,22,30,19)(15,24,28,21)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ | $39$ | $4$ | $( 2, 3)( 4,25,39,18)( 5,27,37,17)( 6,26,38,16)( 7,12,35,32)( 8,11,36,31) ( 9,10,34,33)(13,20,29,23)(14,19,30,22)(15,21,28,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ |
| $ 6, 6, 6, 6, 6, 6, 3 $ | $26$ | $6$ | $( 1, 2, 3)( 4,37, 6,39, 5,38)( 7,36, 9,35, 8,34)(10,31,12,33,11,32) (13,30,15,29,14,28)(16,27,18,26,17,25)(19,23,21,22,20,24)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 4, 9,11,15,18,20,22,25,29,33,34,37, 3, 6, 8,10,14,17,19,24,27,28,32,36, 39, 2, 5, 7,12,13,16,21,23,26,30,31,35,38)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 5, 8,11,13,17,20,23,27,29,31,36,37)( 2, 6, 9,12,14,18,21,24,25,30,32,34, 38)( 3, 4, 7,10,15,16,19,22,26,28,33,35,39)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 6, 7,11,14,16,20,24,26,29,32,35,37, 2, 4, 8,12,15,17,21,22,27,30,33,36, 38, 3, 5, 9,10,13,18,19,23,25,28,31,34,39)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 7,14,20,26,32,37, 4,12,17,22,30,36, 3, 9,13,19,25,31,39, 6,11,16,24,29, 35, 2, 8,15,21,27,33,38, 5,10,18,23,28,34)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1, 8,13,20,27,31,37, 5,11,17,23,29,36)( 2, 9,14,21,25,32,38, 6,12,18,24,30, 34)( 3, 7,15,19,26,33,39, 4,10,16,22,28,35)$ |
| $ 39 $ | $4$ | $39$ | $( 1, 9,15,20,25,33,37, 6,10,17,24,28,36, 2, 7,13,21,26,31,38, 4,11,18,22,29, 34, 3, 8,14,19,27,32,39, 5,12,16,23,30,35)$ |
| $ 13, 13, 13 $ | $4$ | $13$ | $( 1,13,27,37,11,23,36, 8,20,31, 5,17,29)( 2,14,25,38,12,24,34, 9,21,32, 6,18, 30)( 3,15,26,39,10,22,35, 7,19,33, 4,16,28)$ |
| $ 39 $ | $4$ | $39$ | $( 1,14,26,37,12,22,36, 9,19,31, 6,16,29, 2,15,27,38,10,23,34, 7,20,32, 4,17, 30, 3,13,25,39,11,24,35, 8,21,33, 5,18,28)$ |
| $ 39 $ | $4$ | $39$ | $( 1,15,25,37,10,24,36, 7,21,31, 4,18,29, 3,14,27,39,12,23,35, 9,20,33, 6,17, 28, 2,13,26,38,11,22,34, 8,19,32, 5,16,30)$ |
Group invariants
| Order: | $156=2^{2} \cdot 3 \cdot 13$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [156, 10] |
| Character table: |
2 2 2 2 2 1 1 . . . . . . . . .
3 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
1a 2a 4a 4b 3a 6a 39a 13a 39b 39c 13b 39d 13c 39e 39f
2P 1a 1a 2a 2a 3a 3a 39d 13b 39c 39e 13c 39f 13a 39b 39a
3P 1a 2a 4b 4a 1a 2a 13b 13b 13b 13c 13c 13c 13a 13a 13a
5P 1a 2a 4a 4b 3a 6a 39a 13a 39b 39c 13b 39d 13c 39e 39f
7P 1a 2a 4b 4a 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
11P 1a 2a 4b 4a 3a 6a 39d 13b 39c 39e 13c 39f 13a 39b 39a
13P 1a 2a 4a 4b 3a 6a 3a 1a 3a 3a 1a 3a 1a 3a 3a
17P 1a 2a 4a 4b 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
19P 1a 2a 4b 4a 3a 6a 39e 13c 39f 39a 13a 39b 13b 39d 39c
23P 1a 2a 4b 4a 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b
29P 1a 2a 4a 4b 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b
31P 1a 2a 4b 4a 3a 6a 39b 13a 39a 39d 13b 39c 13c 39f 39e
37P 1a 2a 4a 4b 3a 6a 39c 13b 39d 39f 13c 39e 13a 39a 39b
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 -1 A -A 1 -1 1 1 1 1 1 1 1 1 1
X.4 1 -1 -A A 1 -1 1 1 1 1 1 1 1 1 1
X.5 2 -2 . . -1 1 -1 2 -1 -1 2 -1 2 -1 -1
X.6 2 2 . . -1 -1 -1 2 -1 -1 2 -1 2 -1 -1
X.7 4 . . . 4 . B B B D D D C C C
X.8 4 . . . 4 . C C C B B B D D D
X.9 4 . . . 4 . D D D C C C B B B
X.10 4 . . . -2 . E B /E /G D G C /F F
X.11 4 . . . -2 . F C /F /E B E D /G G
X.12 4 . . . -2 . G D /G /F C F B /E E
X.13 4 . . . -2 . /G D G F C /F B E /E
X.14 4 . . . -2 . /E B E G D /G C F /F
X.15 4 . . . -2 . /F C F E B /E D G /G
A = -E(4)
= -Sqrt(-1) = -i
B = E(13)+E(13)^5+E(13)^8+E(13)^12
C = E(13)^4+E(13)^6+E(13)^7+E(13)^9
D = E(13)^2+E(13)^3+E(13)^10+E(13)^11
E = E(39)^23+E(39)^28+E(39)^29+E(39)^37
F = E(39)^14+E(39)^31+E(39)^34+E(39)^38
G = E(39)^7+E(39)^17+E(39)^19+E(39)^35
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