Group action invariants
| Degree $n$ : | $26$ | |
| Transitive number $t$ : | $5$ | |
| Group : | $C_2\times C_{13}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,10,4)(2,9,3)(5,19,16)(6,20,15)(7,12,22)(8,11,21)(17,24,26)(18,23,25), (1,2)(3,7,19,4,8,20)(5,13,11,6,14,12)(9,25,21,10,26,22)(15,17,23,16,18,24) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 39: $C_{13}:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $C_{13}:C_3$
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $13$ | $3$ | $( 3, 8,19)( 4, 7,20)( 5,14,11)( 6,13,12)( 9,26,21)(10,25,22)(15,18,23) (16,17,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $13$ | $3$ | $( 3,19, 8)( 4,20, 7)( 5,11,14)( 6,12,13)( 9,21,26)(10,22,25)(15,23,18) (16,24,17)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$ |
| $ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3, 7,19, 4, 8,20)( 5,13,11, 6,14,12)( 9,25,21,10,26,22) (15,17,23,16,18,24)$ |
| $ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3,20, 8, 4,19, 7)( 5,12,14, 6,11,13)( 9,22,26,10,21,25) (15,24,18,16,23,17)$ |
| $ 26 $ | $3$ | $26$ | $( 1, 3, 6, 8,10,11,13,16,18,19,22,24,25, 2, 4, 5, 7, 9,12,14,15,17,20,21,23,26 )$ |
| $ 13, 13 $ | $3$ | $13$ | $( 1, 4, 6, 7,10,12,13,15,18,20,22,23,25)( 2, 3, 5, 8, 9,11,14,16,17,19,21,24, 26)$ |
| $ 26 $ | $3$ | $26$ | $( 1, 5,10,14,18,21,25, 3, 7,11,15,19,23, 2, 6, 9,13,17,22,26, 4, 8,12,16,20,24 )$ |
| $ 13, 13 $ | $3$ | $13$ | $( 1, 6,10,13,18,22,25, 4, 7,12,15,20,23)( 2, 5, 9,14,17,21,26, 3, 8,11,16,19, 24)$ |
| $ 26 $ | $3$ | $26$ | $( 1, 9,18,26, 7,16,23, 5,13,21, 4,11,20, 2,10,17,25, 8,15,24, 6,14,22, 3,12,19 )$ |
| $ 13, 13 $ | $3$ | $13$ | $( 1,10,18,25, 7,15,23, 6,13,22, 4,12,20)( 2, 9,17,26, 8,16,24, 5,14,21, 3,11, 19)$ |
| $ 13, 13 $ | $3$ | $13$ | $( 1,15, 4,18, 6,20, 7,22,10,23,12,25,13)( 2,16, 3,17, 5,19, 8,21, 9,24,11,26, 14)$ |
| $ 26 $ | $3$ | $26$ | $( 1,16, 4,17, 6,19, 7,21,10,24,12,26,13, 2,15, 3,18, 5,20, 8,22, 9,23,11,25,14 )$ |
Group invariants
| Order: | $78=2 \cdot 3 \cdot 13$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [78, 2] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 1 1 1 1 1 . . . . . . . .
13 1 . . 1 . . 1 1 1 1 1 1 1 1
1a 3a 3b 2a 6a 6b 26a 13a 26b 13b 26c 13c 13d 26d
2P 1a 3b 3a 1a 3b 3a 13b 13b 13c 13c 13d 13d 13a 13a
3P 1a 1a 1a 2a 2a 2a 26a 13a 26b 13b 26c 13c 13d 26d
5P 1a 3b 3a 2a 6b 6a 26b 13b 26c 13c 26d 13d 13a 26a
7P 1a 3a 3b 2a 6a 6b 26d 13d 26a 13a 26b 13b 13c 26c
11P 1a 3b 3a 2a 6b 6a 26d 13d 26a 13a 26b 13b 13c 26c
13P 1a 3a 3b 2a 6a 6b 2a 1a 2a 1a 2a 1a 1a 2a
17P 1a 3b 3a 2a 6b 6a 26c 13c 26d 13d 26a 13a 13b 26b
19P 1a 3a 3b 2a 6a 6b 26b 13b 26c 13c 26d 13d 13a 26a
23P 1a 3b 3a 2a 6b 6a 26c 13c 26d 13d 26a 13a 13b 26b
X.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
X.3 1 A /A -1 -A -/A -1 1 -1 1 -1 1 1 -1
X.4 1 /A A -1 -/A -A -1 1 -1 1 -1 1 1 -1
X.5 1 A /A 1 A /A 1 1 1 1 1 1 1 1
X.6 1 /A A 1 /A A 1 1 1 1 1 1 1 1
X.7 3 . . 3 . . B B /C /C /B /B C C
X.8 3 . . 3 . . C C B B /C /C /B /B
X.9 3 . . 3 . . /C /C /B /B C C B B
X.10 3 . . 3 . . /B /B C C B B /C /C
X.11 3 . . -3 . . -B B -/C /C -/B /B C -C
X.12 3 . . -3 . . -C C -B B -/C /C /B -/B
X.13 3 . . -3 . . -/C /C -/B /B -C C B -B
X.14 3 . . -3 . . -/B /B -C C -B B /C -/C
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(13)^7+E(13)^8+E(13)^11
C = E(13)^4+E(13)^10+E(13)^12
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