Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $22$ | |
| Group : | $D_{21}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,16,9,4,33,42)(2,17,7,6,31,40)(3,18,8,5,32,41)(10,19,23,37,29,26)(11,20,24,38,30,25)(12,21,22,39,28,27)(13,34,15,36,14,35), (1,35,7,12,19,4)(2,34,8,10,20,6)(3,36,9,11,21,5)(13,28,31,40,27,24)(14,30,33,42,25,23)(15,29,32,41,26,22)(16,37,18,39,17,38) | |
| $|\Aut(F/K)|$: | $3$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $S_3$, $C_6$ 18: $S_3\times C_3$ 42: $F_7$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 6: $S_3\times C_3$
Degree 7: $F_7$
Degree 14: $F_7$
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 $ | $14$ | $3$ | $( 4,29,35)( 5,28,36)( 6,30,34)( 7,15,27)( 8,14,26)( 9,13,25)(10,41,18) (11,42,16)(12,40,17)(19,39,32)(20,37,33)(21,38,31)(22,23,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $14$ | $3$ | $( 4,35,29)( 5,36,28)( 6,34,30)( 7,27,15)( 8,26,14)( 9,25,13)(10,18,41) (11,16,42)(12,17,40)(19,32,39)(20,33,37)(21,31,38)(22,24,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,20,21) (22,24,23)(25,27,26)(28,30,29)(31,32,33)(34,35,36)(37,38,39)(40,42,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 2, 3)( 4,36,30)( 5,34,29)( 6,35,28)( 7,26,13)( 8,25,15)( 9,27,14) (10,17,42)(11,18,40)(12,16,41)(19,33,38)(20,31,39)(21,32,37)(22,23,24)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 3, 2)( 4,30,36)( 5,29,34)( 6,28,35)( 7,13,26)( 8,15,25)( 9,14,27) (10,42,17)(11,40,18)(12,41,16)(19,38,33)(20,39,31)(21,37,32)(22,24,23)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 4)( 2, 6)( 3, 5)( 7,40)( 8,41)( 9,42)(10,37)(11,38)(12,39)(13,36)(14,34) (15,35)(16,33)(17,31)(18,32)(19,29)(20,30)(21,28)(22,27)(23,26)(24,25)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,15,40,39,29)( 2, 6,14,41,37,30)( 3, 5,13,42,38,28)( 7,22,25,36,21,17) ( 8,23,27,35,19,18)( 9,24,26,34,20,16)(10,32,12,31,11,33)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $21$ | $6$ | $( 1, 4,19,12, 7,35)( 2, 6,20,10, 8,34)( 3, 5,21,11, 9,36)(13,24,27,40,31,28) (14,23,25,42,33,30)(15,22,26,41,32,29)(16,38,17,39,18,37)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1, 7,13,21,27,32,38, 3, 9,14,20,25,31,37, 2, 8,15,19,26,33,39) ( 4,12,16,23,29,35,41, 6,10,17,24,30,34,42, 5,11,18,22,28,36,40)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 8,14,21,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,20,27,33,37) ( 4,11,17,23,28,34,41)( 5,10,16,22,30,35,40)( 6,12,18,24,29,36,42)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1, 9,15,21,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,20,26,32,37) ( 4,10,18,23,30,36,41, 5,12,17,22,29,34,40, 6,11,16,24,28,35,42)$ |
Group invariants
| Order: | $126=2 \cdot 3^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [126, 9] |
| Character table: |
2 1 . . . 1 1 1 1 1 . . .
3 2 2 2 2 2 2 1 1 1 1 1 1
7 1 . . 1 . . . . . 1 1 1
1a 3a 3b 3c 3d 3e 2a 6a 6b 21a 7a 21b
2P 1a 3b 3a 3c 3e 3d 1a 3e 3d 21b 7a 21a
3P 1a 1a 1a 1a 1a 1a 2a 2a 2a 7a 7a 7a
5P 1a 3b 3a 3c 3e 3d 2a 6b 6a 21a 7a 21b
7P 1a 3a 3b 3c 3d 3e 2a 6a 6b 3c 1a 3c
11P 1a 3b 3a 3c 3e 3d 2a 6b 6a 21b 7a 21a
13P 1a 3a 3b 3c 3d 3e 2a 6a 6b 21b 7a 21a
17P 1a 3b 3a 3c 3e 3d 2a 6b 6a 21a 7a 21b
19P 1a 3a 3b 3c 3d 3e 2a 6a 6b 21b 7a 21a
X.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
X.3 1 A /A 1 /A A -1 -/A -A 1 1 1
X.4 1 /A A 1 A /A -1 -A -/A 1 1 1
X.5 1 A /A 1 /A A 1 /A A 1 1 1
X.6 1 /A A 1 A /A 1 A /A 1 1 1
X.7 2 -1 -1 -1 2 2 . . . -1 2 -1
X.8 2 -/A -A -1 B /B . . . -1 2 -1
X.9 2 -A -/A -1 /B B . . . -1 2 -1
X.10 6 . . 6 . . . . . -1 -1 -1
X.11 6 . . -3 . . . . . C -1 *C
X.12 6 . . -3 . . . . . *C -1 C
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
= -1-Sqrt(-3) = -1-i3
C = E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19
= (1-Sqrt(21))/2 = -b21
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