Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $23$ | |
| Group : | $S_3\times C_7:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,9,20)(2,8,21)(3,7,19)(4,35,12)(5,36,10)(6,34,11)(13,31,27)(14,32,26)(15,33,25)(16,17,18)(22,29,40)(23,30,42)(24,28,41), (1,18,15,24,37,36)(2,16,14,22,38,34)(3,17,13,23,39,35)(4,8,41,26,28,19)(5,7,40,27,29,20)(6,9,42,25,30,21)(10,31,12,32,11,33) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ 21: $C_7:C_3$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 7: $C_7:C_3$
Degree 14: $(C_7:C_3) \times C_2$
Degree 21: None
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $14$ | $3$ | $( 4,28,36)( 5,29,34)( 6,30,35)( 7,15,25)( 8,13,27)( 9,14,26)(10,42,16) (11,41,17)(12,40,18)(19,38,33)(20,39,32)(21,37,31)(22,23,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $14$ | $3$ | $( 4,36,28)( 5,34,29)( 6,35,30)( 7,25,15)( 8,27,13)( 9,26,14)(10,16,42) (11,17,41)(12,18,40)(19,33,38)(20,32,39)(21,31,37)(22,24,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,20,21) (22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,38,39)(40,41,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 2, 3)( 4,35,29)( 5,36,30)( 6,34,28)( 7,26,13)( 8,25,14)( 9,27,15) (10,18,41)(11,16,40)(12,17,42)(19,32,37)(20,31,38)(21,33,39)(22,23,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 2)( 4,29,35)( 5,30,36)( 6,28,34)( 7,13,26)( 8,14,25)( 9,15,27) (10,41,18)(11,40,16)(12,42,17)(19,37,32)(20,38,31)(21,39,33)(22,24,23)$ |
| $ 14, 14, 14 $ | $9$ | $14$ | $( 1, 4, 9,10,14,17,20,24,26,30,32,34,39,40)( 2, 5, 8,11,13,18,21,22,27,28,31, 35,37,42)( 3, 6, 7,12,15,16,19,23,25,29,33,36,38,41)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,31,22,25,12)( 2, 5,33,23,26,10)( 3, 6,32,24,27,11)( 7,16,14,30,37,35) ( 8,18,15,29,39,34)( 9,17,13,28,38,36)(19,41,20,40,21,42)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,38,23,27,18)( 2, 5,39,24,25,16)( 3, 6,37,22,26,17)( 7,29, 8,28, 9,30) (10,19,36,31,42,14)(11,20,34,33,41,13)(12,21,35,32,40,15)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1, 7,13,20,25,31,39, 3, 8,14,19,27,32,38, 2, 9,15,21,26,33,37) ( 4,11,16,24,28,36,40, 5,12,17,22,29,34,42, 6,10,18,23,30,35,41)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 9,14,20,26,32,39)( 2, 8,13,21,27,31,37)( 3, 7,15,19,25,33,38) ( 4,10,17,24,30,34,40)( 5,11,18,22,28,35,42)( 6,12,16,23,29,36,41)$ |
| $ 14, 14, 14 $ | $9$ | $14$ | $( 1,10,20,30,39, 4,14,24,32,40, 9,17,26,34)( 2,11,21,28,37, 5,13,22,31,42, 8, 18,27,35)( 3,12,19,29,38, 6,15,23,33,41, 7,16,25,36)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1,19,37,14,33, 8,26, 3,21,39,15,31, 9,25, 2,20,38,13,32, 7,27) ( 4,22,41,17,35,12,30, 5,23,40,18,36,10,28, 6,24,42,16,34,11,29)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,20,39,14,32, 9,26)( 2,21,37,13,31, 8,27)( 3,19,38,15,33, 7,25) ( 4,24,40,17,34,10,30)( 5,22,42,18,35,11,28)( 6,23,41,16,36,12,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,30)( 8,29)( 9,28)(10,33)(11,32) (12,31)(13,36)(14,35)(15,34)(16,37)(17,38)(18,39)(19,40)(20,42)(21,41)$ |
Group invariants
| Order: | $126=2 \cdot 3^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [126, 8] |
| Character table: |
2 1 . . . 1 1 1 1 1 . 1 1 . 1 1
3 2 2 2 2 2 2 . 1 1 1 1 . 1 1 1
7 1 . . 1 . . 1 . . 1 1 1 1 1 1
1a 3a 3b 3c 3d 3e 14a 6a 6b 21a 7a 14b 21b 7b 2a
2P 1a 3b 3a 3c 3e 3d 7a 3d 3e 21a 7a 7b 21b 7b 1a
3P 1a 1a 1a 1a 1a 1a 14b 2a 2a 7b 7b 14a 7a 7a 2a
5P 1a 3b 3a 3c 3e 3d 14b 6b 6a 21b 7b 14a 21a 7a 2a
7P 1a 3a 3b 3c 3d 3e 2a 6a 6b 3c 1a 2a 3c 1a 2a
11P 1a 3b 3a 3c 3e 3d 14a 6b 6a 21a 7a 14b 21b 7b 2a
13P 1a 3a 3b 3c 3d 3e 14b 6a 6b 21b 7b 14a 21a 7a 2a
17P 1a 3b 3a 3c 3e 3d 14b 6b 6a 21b 7b 14a 21a 7a 2a
19P 1a 3a 3b 3c 3d 3e 14b 6a 6b 21b 7b 14a 21a 7a 2a
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 1 /A A -1 -A -/A 1 1 -1 1 1 -1
X.4 1 /A A 1 A /A -1 -/A -A 1 1 -1 1 1 -1
X.5 1 A /A 1 /A A 1 A /A 1 1 1 1 1 1
X.6 1 /A A 1 A /A 1 /A A 1 1 1 1 1 1
X.7 2 -1 -1 -1 2 2 . . . -1 2 . -1 2 .
X.8 2 -/A -A -1 B /B . . . -1 2 . -1 2 .
X.9 2 -A -/A -1 /B B . . . -1 2 . -1 2 .
X.10 3 . . 3 . . C . . -C -C /C -/C -/C -3
X.11 3 . . 3 . . /C . . -/C -/C C -C -C -3
X.12 3 . . 3 . . -/C . . -/C -/C -C -C -C 3
X.13 3 . . 3 . . -C . . -C -C -/C -/C -/C 3
X.14 6 . . -3 . . . . . /C D . C /D .
X.15 6 . . -3 . . . . . C /D . /C D .
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
C = -E(7)-E(7)^2-E(7)^4
= (1-Sqrt(-7))/2 = -b7
D = 2*E(7)^3+2*E(7)^5+2*E(7)^6
= -1-Sqrt(-7) = -1-i7
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