Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $18$ | |
| Group : | $D_{21}:C_3$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,26,41,12,21,34,4,13,28,37,7,23,36,6,16,29,40,9,19,32)(2,18,25,42,11,22,33,3,14,27,38,8,24,35,5,15,30,39,10,20,31), (1,33,41,15,28,24)(2,34,42,16,27,23)(3,32,38,13,30,21)(4,31,37,14,29,22)(5,36,39,17,25,19)(6,35,40,18,26,20)(7,11)(8,12)(9,10) | |
| $|\Aut(F/K)|$: | $6$ |
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: $S_3$
Degree 6: $S_3$
Degree 7: $F_7$
Degree 14: $F_7$
Degree 21: 21T10
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, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,13,26)( 8,14,25)( 9,16,28)(10,15,27)(11,18,30)(12,17,29)(19,37,36) (20,38,35)(21,40,32)(22,39,31)(23,41,34)(24,42,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,26,13)( 8,25,14)( 9,28,16)(10,27,15)(11,30,18)(12,29,17)(19,36,37) (20,35,38)(21,32,40)(22,31,39)(23,34,41)(24,33,42)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,20,13,38,26,35)( 8,19,14,37,25,36)( 9,24,16,42,28,33) (10,23,15,41,27,34)(11,21,18,40,30,32)(12,22,17,39,29,31)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 6)( 4, 5)( 7,35,26,38,13,20)( 8,36,25,37,14,19)( 9,33,28,42,16,24) (10,34,27,41,15,23)(11,32,30,40,18,21)(12,31,29,39,17,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $21$ | $2$ | $( 1, 2)( 3, 6)( 4, 5)( 7,38)( 8,37)( 9,42)(10,41)(11,40)(12,39)(13,35)(14,36) (15,34)(16,33)(17,31)(18,32)(19,25)(20,26)(21,30)(22,29)(23,27)(24,28)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(13,16,17)(14,15,18)(19,21,23) (20,22,24)(25,27,30)(26,28,29)(31,33,35)(32,34,36)(37,40,41)(38,39,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7,16,29)( 8,15,30)( 9,17,26)(10,18,25)(11,14,27) (12,13,28)(19,40,34)(20,39,33)(21,41,36)(22,42,35)(23,37,32)(24,38,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 4, 6)( 2, 3, 5)( 7,28,17)( 8,27,18)( 9,29,13)(10,30,14)(11,25,15) (12,26,16)(19,32,41)(20,31,42)(21,34,37)(22,33,38)(23,36,40)(24,35,39)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1, 7,17,23,26,36,41, 6,12,16,21,29,34,40, 4, 9,13,19,28,32,37) ( 2, 8,18,24,25,35,42, 5,11,15,22,30,33,39, 3,10,14,20,27,31,38)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 9,16,23,28,34,41)( 2,10,15,24,27,33,42)( 3,11,18,20,30,35,38) ( 4,12,17,19,29,36,37)( 5, 8,14,22,25,31,39)( 6, 7,13,21,26,32,40)$ |
| $ 21, 21 $ | $6$ | $21$ | $( 1,12,13,23,29,32,41, 4, 7,16,19,26,34,37, 6, 9,17,21,28,36,40) ( 2,11,14,24,30,31,42, 3, 8,15,20,25,33,38, 5,10,18,22,27,35,39)$ |
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 1 1 1 2 2 2 1 1 1
7 1 . . . . . 1 . . 1 1 1
1a 3a 3b 6a 6b 2a 3c 3d 3e 21a 7a 21b
2P 1a 3b 3a 3a 3b 1a 3c 3e 3d 21b 7a 21a
3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 7a 7a 7a
5P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21a 7a 21b
7P 1a 3a 3b 6a 6b 2a 3c 3d 3e 3c 1a 3c
11P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21b 7a 21a
13P 1a 3a 3b 6a 6b 2a 3c 3d 3e 21b 7a 21a
17P 1a 3b 3a 6b 6a 2a 3c 3e 3d 21a 7a 21b
19P 1a 3a 3b 6a 6b 2a 3c 3d 3e 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 -/A -A -1 1 A /A 1 1 1
X.4 1 /A A -A -/A -1 1 /A A 1 1 1
X.5 1 A /A /A A 1 1 A /A 1 1 1
X.6 1 /A A A /A 1 1 /A A 1 1 1
X.7 2 2 2 . . . -1 -1 -1 -1 2 -1
X.8 2 B /B . . . -1 -A -/A -1 2 -1
X.9 2 /B B . . . -1 -/A -A -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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