Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $39$ | |
| Group : | $A_4\times C_7:C_3$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,14,25,39,9,23,36,3,16,28,41,12,20,32,5,18,30,38,7,21,33)(2,13,26,40,10,24,35,4,15,27,42,11,19,31,6,17,29,37,8,22,34), (1,21,15)(2,22,16)(3,24,18)(4,23,17)(5,19,13)(6,20,14)(7,32,37)(8,31,38)(9,34,39)(10,33,40)(11,36,41)(12,35,42)(25,27,29)(26,28,30) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ x 4 9: $C_3^2$ 12: $A_4$ 21: $C_7:C_3$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4$
Degree 7: $C_7:C_3$
Degree 14: None
Degree 21: 21T7
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 $ | $7$ | $3$ | $( 7,18,28)( 8,17,27)( 9,14,30)(10,13,29)(11,15,26)(12,16,25)(19,40,35) (20,39,36)(21,41,32)(22,42,31)(23,38,33)(24,37,34)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,28,18)( 8,27,17)( 9,30,14)(10,29,13)(11,26,15)(12,25,16)(19,35,40) (20,36,39)(21,32,41)(22,31,42)(23,33,38)(24,34,37)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(15,16)(21,22)(23,24)(25,26)(29,30)(31,32) (33,34)(37,38)(41,42)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 4)( 5, 6)( 7,18,28)( 8,17,27)( 9,13,30,10,14,29)(11,16,26,12,15,25) (19,40,35)(20,39,36)(21,42,32,22,41,31)(23,37,33,24,38,34)$ |
| $ 6, 6, 6, 6, 3, 3, 3, 3, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 4)( 5, 6)( 7,28,18)( 8,27,17)( 9,29,14,10,30,13)(11,25,15,12,26,16) (19,35,40)(20,36,39)(21,31,41,22,32,42)(23,34,38,24,33,37)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,15,17)(14,16,18)(19,22,24) (20,21,23)(25,28,30)(26,27,29)(31,34,35)(32,33,36)(37,40,42)(38,39,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,14,25)( 8,13,26)( 9,16,28)(10,15,27)(11,17,29) (12,18,30)(19,42,34)(20,41,33)(21,38,36)(22,37,35)(23,39,32)(24,40,31)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,30,16)( 8,29,15)( 9,25,18)(10,26,17)(11,27,13) (12,28,14)(19,31,37)(20,32,38)(21,33,39)(22,34,40)(23,36,41)(24,35,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,17,15)(14,18,16)(19,24,22) (20,23,21)(25,30,28)(26,29,27)(31,35,34)(32,36,33)(37,42,40)(38,41,39)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,16,30)( 8,15,29)( 9,18,25)(10,17,26)(11,13,27) (12,14,28)(19,37,31)(20,38,32)(21,39,33)(22,40,34)(23,41,36)(24,42,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,14)( 8,26,13)( 9,28,16)(10,27,15)(11,29,17) (12,30,18)(19,34,42)(20,33,41)(21,36,38)(22,35,37)(23,32,39)(24,31,40)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 7,18,20,28,36,39)( 2, 8,17,19,27,35,40)( 3, 9,14,21,30,32,41) ( 4,10,13,22,29,31,42)( 5,12,16,23,25,33,38)( 6,11,15,24,26,34,37)$ |
| $ 14, 14, 7, 7 $ | $9$ | $14$ | $( 1, 7,18,20,28,36,39)( 2, 8,17,19,27,35,40)( 3,10,14,22,30,31,41, 4, 9,13,21, 29,32,42)( 5,11,16,24,25,34,38, 6,12,15,23,26,33,37)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1, 9,15,20,30,34,39, 3,11,18,21,26,36,41, 6, 7,14,24,28,32,37) ( 2,10,16,19,29,33,40, 4,12,17,22,25,35,42, 5, 8,13,23,27,31,38)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1,11,14,20,26,32,39, 6, 9,18,24,30,36,37, 3, 7,15,21,28,34,41) ( 2,12,13,19,25,31,40, 5,10,17,23,29,35,38, 4, 8,16,22,27,33,42)$ |
| $ 14, 14, 7, 7 $ | $9$ | $14$ | $( 1,19,39,17,36, 8,28, 2,20,40,18,35, 7,27)( 3,21,41,14,32, 9,30) ( 4,22,42,13,31,10,29)( 5,24,38,15,33,11,25, 6,23,37,16,34,12,26)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,20,39,18,36, 7,28)( 2,19,40,17,35, 8,27)( 3,21,41,14,32, 9,30) ( 4,22,42,13,31,10,29)( 5,23,38,16,33,12,25)( 6,24,37,15,34,11,26)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1,21,38,18,32,12,28, 3,23,39,14,33, 7,30, 5,20,41,16,36, 9,25) ( 2,22,37,17,31,11,27, 4,24,40,13,34, 8,29, 6,19,42,15,35,10,26)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1,23,42,18,33,10,28, 5,22,39,16,31, 7,25, 4,20,38,13,36,12,29) ( 2,24,41,17,34, 9,27, 6,21,40,15,32, 8,26, 3,19,37,14,35,11,30)$ |
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [252, 27] |
| Character table: |
2 2 2 2 2 2 2 . . . . . . 2 2 . . 2 2 . .
3 2 2 2 1 1 1 2 2 2 2 2 2 1 . 1 1 . 1 1 1
7 1 . . 1 . . 1 . . 1 . . 1 1 1 1 1 1 1 1
1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7a 14a 21a 21b 14b 7b 21c 21d
2P 1a 3b 3a 1a 3b 3a 3f 3h 3g 3c 3e 3d 7a 7a 21b 21a 7b 7b 21d 21c
3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 1a 1a 1a 7b 14b 7b 7b 14a 7a 7a 7a
5P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a
7P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 1a 2a 3c 3f 2a 1a 3c 3f
11P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7a 14a 21b 21a 14b 7b 21d 21c
13P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b
17P 1a 3b 3a 2a 6b 6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a
19P 1a 3a 3b 2a 6a 6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b
X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 1 1 1 1 A A A /A /A /A 1 1 A /A 1 1 A /A
X.3 1 1 1 1 1 1 /A /A /A A A A 1 1 /A A 1 1 /A A
X.4 1 A /A 1 A /A 1 A /A 1 A /A 1 1 1 1 1 1 1 1
X.5 1 /A A 1 /A A 1 /A A 1 /A A 1 1 1 1 1 1 1 1
X.6 1 A /A 1 A /A A /A 1 /A 1 A 1 1 A /A 1 1 A /A
X.7 1 /A A 1 /A A /A A 1 A 1 /A 1 1 /A A 1 1 /A A
X.8 1 A /A 1 A /A /A 1 A A /A 1 1 1 /A A 1 1 /A A
X.9 1 /A A 1 /A A A 1 /A /A A 1 1 1 A /A 1 1 A /A
X.10 3 3 3 -1 -1 -1 . . . . . . 3 -1 . . -1 3 . .
X.11 3 . . 3 . . 3 . . 3 . . C C C C /C /C /C /C
X.12 3 . . 3 . . 3 . . 3 . . /C /C /C /C C C C C
X.13 3 . . 3 . . B . . /B . . C C E F /C /C /F /E
X.14 3 . . 3 . . /B . . B . . C C F E /C /C /E /F
X.15 3 . . 3 . . B . . /B . . /C /C /F /E C C E F
X.16 3 . . 3 . . /B . . B . . /C /C /E /F C C F E
X.17 3 B /B -1 -A -/A . . . . . . 3 -1 . . -1 3 . .
X.18 3 /B B -1 -/A -A . . . . . . 3 -1 . . -1 3 . .
X.19 9 . . -3 . . . . . . . . D -C . . -/C /D . .
X.20 9 . . -3 . . . . . . . . /D -/C . . -C D . .
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
= (-3-3*Sqrt(-3))/2 = -3-3b3
C = E(7)^3+E(7)^5+E(7)^6
= (-1-Sqrt(-7))/2 = -1-b7
D = 3*E(7)^3+3*E(7)^5+3*E(7)^6
= (-3-3*Sqrt(-7))/2 = -3-3b7
E = E(21)^2+E(21)^8+E(21)^11
F = E(21)+E(21)^4+E(21)^16
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