Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $2$ | |
| Group : | $C_2\times C_7:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,18,24,2,17,23)(3,14,19,4,13,20)(5,16,21,6,15,22)(7,41,32,8,42,31)(9,37,33,10,38,34)(11,40,35,12,39,36)(25,30,27,26,29,28), (1,7,13,22,28,36,38,2,8,14,21,27,35,37)(3,10,15,23,30,32,39,4,9,16,24,29,31,40)(5,12,17,20,26,34,41,6,11,18,19,25,33,42) | |
| $|\Aut(F/K)|$: | $42$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 21: $C_7:C_3$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$
Degree 21: 21T2
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
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, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,16,25)( 8,15,26)( 9,17,28)(10,18,27)(11,13,30) (12,14,29)(19,38,31)(20,37,32)(21,39,33)(22,40,34)(23,42,36)(24,41,35)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 4, 5, 2, 3, 6)( 7,15,25, 8,16,26)( 9,18,28,10,17,27)(11,14,30,12,13,29) (19,37,31,20,38,32)(21,40,33,22,39,34)(23,41,36,24,42,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,16)( 8,26,15)( 9,28,17)(10,27,18)(11,30,13) (12,29,14)(19,31,38)(20,32,37)(21,33,39)(22,34,40)(23,36,42)(24,35,41)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 6, 3, 2, 5, 4)( 7,26,16, 8,25,15)( 9,27,17,10,28,18)(11,29,13,12,30,14) (19,32,38,20,31,37)(21,34,39,22,33,40)(23,35,42,24,36,41)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1, 7,13,22,28,36,38, 2, 8,14,21,27,35,37)( 3,10,15,23,30,32,39, 4, 9,16,24, 29,31,40)( 5,12,17,20,26,34,41, 6,11,18,19,25,33,42)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 8,13,21,28,35,38)( 2, 7,14,22,27,36,37)( 3, 9,15,24,30,31,39) ( 4,10,16,23,29,32,40)( 5,11,17,19,26,33,41)( 6,12,18,20,25,34,42)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,21,38,13,35, 8,28)( 2,22,37,14,36, 7,27)( 3,24,39,15,31, 9,30) ( 4,23,40,16,32,10,29)( 5,19,41,17,33,11,26)( 6,20,42,18,34,12,25)$ |
| $ 14, 14, 14 $ | $3$ | $14$ | $( 1,22,38,14,35, 7,28, 2,21,37,13,36, 8,27)( 3,23,39,16,31,10,30, 4,24,40,15, 32, 9,29)( 5,20,41,18,33,12,26, 6,19,42,17,34,11,25)$ |
Group invariants
| Order: | $42=2 \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [42, 2] |
| Character table: |
2 1 1 1 1 1 1 1 1 1 1
3 1 1 1 1 1 1 . . . .
7 1 1 . . . . 1 1 1 1
1a 2a 3a 6a 3b 6b 14a 7a 7b 14b
2P 1a 1a 3b 3b 3a 3a 7a 7a 7b 7b
3P 1a 2a 1a 2a 1a 2a 14b 7b 7a 14a
5P 1a 2a 3b 6b 3a 6a 14b 7b 7a 14a
7P 1a 2a 3a 6a 3b 6b 2a 1a 1a 2a
11P 1a 2a 3b 6b 3a 6a 14a 7a 7b 14b
13P 1a 2a 3a 6a 3b 6b 14b 7b 7a 14a
X.1 1 1 1 1 1 1 1 1 1 1
X.2 1 -1 1 -1 1 -1 -1 1 1 -1
X.3 1 -1 A -A /A -/A -1 1 1 -1
X.4 1 -1 /A -/A A -A -1 1 1 -1
X.5 1 1 A A /A /A 1 1 1 1
X.6 1 1 /A /A A A 1 1 1 1
X.7 3 -3 . . . . B -B -/B /B
X.8 3 -3 . . . . /B -/B -B B
X.9 3 3 . . . . -/B -/B -B -B
X.10 3 3 . . . . -B -B -/B -/B
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = -E(7)-E(7)^2-E(7)^4
= (1-Sqrt(-7))/2 = -b7
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