Group action invariants
| Degree $n$ : | $39$ | |
| Transitive number $t$ : | $37$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,37,30,2,39,28)(3,38,29)(4,8,16)(5,7,17,6,9,18)(10,26,32,11,25,33)(12,27,31)(13,36,19,14,35,20)(15,34,21)(22,24), (1,33,6,3,32,5,2,31,4)(7,12,20,8,10,21,9,11,19)(13,29,34)(14,30,35)(15,28,36)(16,38,23,18,37,22,17,39,24)(25,26,27) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 39: $C_{13}:C_3$ 78: 26T5 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $C_{13}:C_3$
Low degree siblings
27T422Siblings 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 $ | $26$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(16,17,18)(19,21,20)(22,23,24) (31,32,33)(37,38,39)$ |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,36,29,22,16,10, 6,37,32,25,21,13, 9)( 2,34,30,23,17,11, 4,38,33,26,19,14, 7)( 3,35,28,24,18,12, 5,39,31,27,20,15, 8)$ |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,29,16, 6,32,21, 9,36,22,10,37,25,13)( 2,30,17, 4,33,19, 7,34,23,11,38,26, 14)( 3,28,18, 5,31,20, 8,35,24,12,39,27,15)$ |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,16,32, 9,22,37,13,29, 6,21,36,10,25)( 2,17,33, 7,23,38,14,30, 4,19,34,11, 26)( 3,18,31, 8,24,39,15,28, 5,20,35,12,27)$ |
| $ 13, 13, 13 $ | $81$ | $13$ | $( 1,32,22,13, 6,36,25,16, 9,37,29,21,10)( 2,33,23,14, 4,34,26,17, 7,38,30,19, 11)( 3,31,24,15, 5,35,27,18, 8,39,28,20,12)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $117$ | $3$ | $( 4,29,12)( 5,30,10)( 6,28,11)( 7,18,20)( 8,16,21)( 9,17,19)(13,32,39) (14,33,37)(15,31,38)(22,36,26)(23,34,27)(24,35,25)$ |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1, 2, 3)( 4,29,11)( 5,30,12)( 6,28,10)( 7,16,20, 9,18,19, 8,17,21) (13,33,38,15,32,37,14,31,39)(22,36,26,23,34,27,24,35,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $117$ | $3$ | $( 4,12,29)( 5,10,30)( 6,11,28)( 7,20,18)( 8,21,16)( 9,19,17)(13,39,32) (14,37,33)(15,38,31)(22,26,36)(23,27,34)(24,25,35)$ |
| $ 9, 9, 9, 3, 3, 3, 3 $ | $234$ | $9$ | $( 1, 2, 3)( 4,11,28)( 5,12,29)( 6,10,30)( 7,19,18, 9,21,17, 8,20,16) (13,37,31,15,39,33,14,38,32)(22,26,36,23,27,34,24,25,35)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(13,15)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32) (34,35)(37,38)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,36,29,23,18,11, 5,37,33,26,21,13, 8)( 2,35,30,22,16,10, 6,39,31,25,19,15, 9, 3,34,28,24,17,12, 4,38,32,27,20,14, 7)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,29,17, 5,32,20, 7,34,24,11,39,27,15)( 2,28,18, 4,33,19, 8,36,22,10,37,26, 13, 3,30,16, 6,31,21, 9,35,23,12,38,25,14)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,16,32, 8,22,37,14,30, 6,20,34,12,26)( 2,18,33, 7,23,39,15,29, 4,19,35,11, 27, 3,17,31, 9,24,38,13,28, 5,21,36,10,25)$ |
| $ 26, 13 $ | $81$ | $26$ | $( 1,32,24,14, 4,34,27,18, 7,38,29,19,12, 2,31,22,13, 5,36,25,17, 8,37,30,21,10 )( 3,33,23,15, 6,35,26,16, 9,39,28,20,11)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $351$ | $6$ | $( 2, 3)( 4,29,10, 5,28,11)( 6,30,12)( 7,18,19)( 8,17,20, 9,16,21)(13,31,39) (14,33,37,15,32,38)(22,36,26,24,34,25)(23,35,27)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ | $351$ | $6$ | $( 2, 3)( 4,12,28, 6,10,30)( 5,11,29)( 7,20,17)( 8,19,18, 9,21,16) (13,38,33,14,37,31)(15,39,32)(22,26,35,23,25,36)(24,27,34)$ |
Group invariants
| Order: | $2106=2 \cdot 3^{4} \cdot 13$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: |
2 1 . 1 1 1 1 1 . 1 . 1 1 1 1 1 1 1
3 4 4 . . . . 2 2 2 2 1 . . . . 1 1
13 1 . 1 1 1 1 . . . . 1 1 1 1 1 . .
1a 3a 13a 13b 13c 13d 3b 9a 3c 9b 2a 26a 26b 26c 26d 6a 6b
2P 1a 3a 13b 13c 13d 13a 3c 9b 3b 9a 1a 13b 13c 13d 13a 3c 3b
3P 1a 1a 13a 13b 13c 13d 1a 3a 1a 3a 2a 26a 26b 26c 26d 2a 2a
5P 1a 3a 13b 13c 13d 13a 3c 9b 3b 9a 2a 26b 26c 26d 26a 6b 6a
7P 1a 3a 13d 13a 13b 13c 3b 9a 3c 9b 2a 26d 26a 26b 26c 6a 6b
11P 1a 3a 13d 13a 13b 13c 3c 9b 3b 9a 2a 26d 26a 26b 26c 6b 6a
13P 1a 3a 1a 1a 1a 1a 3b 9a 3c 9b 2a 2a 2a 2a 2a 6a 6b
17P 1a 3a 13c 13d 13a 13b 3c 9b 3b 9a 2a 26c 26d 26a 26b 6b 6a
19P 1a 3a 13b 13c 13d 13a 3b 9a 3c 9b 2a 26b 26c 26d 26a 6a 6b
23P 1a 3a 13c 13d 13a 13b 3c 9b 3b 9a 2a 26c 26d 26a 26b 6b 6a
X.1 1 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 -1 -1
X.3 1 1 1 1 1 1 C C /C /C -1 -1 -1 -1 -1 -C -/C
X.4 1 1 1 1 1 1 /C /C C C -1 -1 -1 -1 -1 -/C -C
X.5 1 1 1 1 1 1 C C /C /C 1 1 1 1 1 C /C
X.6 1 1 1 1 1 1 /C /C C C 1 1 1 1 1 /C C
X.7 3 3 A B /A /B . . . . 3 A B /A /B . .
X.8 3 3 /A /B A B . . . . 3 /A /B A B . .
X.9 3 3 B /A /B A . . . . 3 B /A /B A . .
X.10 3 3 /B A B /A . . . . 3 /B A B /A . .
X.11 3 3 A B /A /B . . . . -3 -A -B -/A -/B . .
X.12 3 3 /A /B A B . . . . -3 -/A -/B -A -B . .
X.13 3 3 B /A /B A . . . . -3 -B -/A -/B -A . .
X.14 3 3 /B A B /A . . . . -3 -/B -A -B -/A . .
X.15 26 -1 . . . . 2 -1 2 -1 . . . . . . .
X.16 26 -1 . . . . D -C /D -/C . . . . . . .
X.17 26 -1 . . . . /D -/C D -C . . . . . . .
A = E(13)+E(13)^3+E(13)^9
B = E(13)^2+E(13)^5+E(13)^6
C = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
D = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
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