Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $8$ | |
| Group : | $C_7:A_4$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,12,31)(2,11,32)(3,8,34)(4,7,33)(5,9,35)(6,10,36)(13,17,15)(14,18,16)(19,39,28)(20,40,27)(21,41,30)(22,42,29)(23,37,26)(24,38,25), (1,16,42)(2,15,41)(3,17,38)(4,18,37)(5,14,40)(6,13,39)(7,30,20)(8,29,19)(9,25,22)(10,26,21)(11,28,24)(12,27,23)(31,34,35)(32,33,36) | |
| $|\Aut(F/K)|$: | $14$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 3: $C_3$ 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: 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32) (33,34)(39,40)(41,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,16,26)( 8,15,25)( 9,18,27)(10,17,28)(11,13,30) (12,14,29)(19,38,32)(20,37,31)(21,40,33)(22,39,34)(23,41,36)(24,42,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,26,16)( 8,25,15)( 9,27,18)(10,28,17)(11,30,13) (12,29,14)(19,32,38)(20,31,37)(21,33,40)(22,34,39)(23,36,41)(24,35,42)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1, 7,13,22,27,36,38)( 2, 8,14,21,28,35,37)( 3, 9,16,23,30,32,39) ( 4,10,15,24,29,31,40)( 5,11,18,19,26,34,41)( 6,12,17,20,25,33,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 7,13,22,27,36,38)( 2, 8,14,21,28,35,37)( 3,10,16,24,30,31,39, 4, 9,15,23, 29,32,40)( 5,12,18,20,26,33,41, 6,11,17,19,25,34,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 8,13,21,27,35,38, 2, 7,14,22,28,36,37)( 3, 9,16,23,30,32,39) ( 4,10,15,24,29,31,40)( 5,12,18,20,26,33,41, 6,11,17,19,25,34,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1, 8,13,21,27,35,38, 2, 7,14,22,28,36,37)( 3,10,16,24,30,31,39, 4, 9,15,23, 29,32,40)( 5,11,18,19,26,34,41)( 6,12,17,20,25,33,42)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,21,38,14,36, 8,27, 2,22,37,13,35, 7,28)( 3,23,39,16,32, 9,30) ( 4,24,40,15,31,10,29)( 5,20,41,17,34,12,26, 6,19,42,18,33,11,25)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,21,38,14,36, 8,27, 2,22,37,13,35, 7,28)( 3,24,39,15,32,10,30, 4,23,40,16, 31, 9,29)( 5,19,41,18,34,11,26)( 6,20,42,17,33,12,25)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $3$ | $7$ | $( 1,22,38,13,36, 7,27)( 2,21,37,14,35, 8,28)( 3,23,39,16,32, 9,30) ( 4,24,40,15,31,10,29)( 5,19,41,18,34,11,26)( 6,20,42,17,33,12,25)$ |
| $ 14, 14, 7, 7 $ | $3$ | $14$ | $( 1,22,38,13,36, 7,27)( 2,21,37,14,35, 8,28)( 3,24,39,15,32,10,30, 4,23,40,16, 31, 9,29)( 5,20,41,17,34,12,26, 6,19,42,18,33,11,25)$ |
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [84, 11] |
| Character table: |
2 2 2 . . 2 2 2 2 2 2 2 2
3 1 . 1 1 . . . . . . . .
7 1 1 . . 1 1 1 1 1 1 1 1
1a 2a 3a 3b 7a 14a 14b 14c 14d 14e 7b 14f
2P 1a 1a 3b 3a 7a 7a 7a 7a 7b 7b 7b 7b
3P 1a 2a 1a 1a 7b 14f 14d 14e 14a 14b 7a 14c
5P 1a 2a 3b 3a 7b 14d 14e 14f 14b 14c 7a 14a
7P 1a 2a 3a 3b 1a 2a 2a 2a 2a 2a 1a 2a
11P 1a 2a 3b 3a 7a 14b 14c 14a 14e 14f 7b 14d
13P 1a 2a 3a 3b 7b 14e 14f 14d 14c 14a 7a 14b
X.1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 1 1 A /A 1 1 1 1 1 1 1 1
X.3 1 1 /A A 1 1 1 1 1 1 1 1
X.4 3 -1 . . 3 -1 -1 -1 -1 -1 3 -1
X.5 3 3 . . B B B B /B /B /B /B
X.6 3 3 . . /B /B /B /B B B B B
X.7 3 -1 . . B C D E /E /C /B /D
X.8 3 -1 . . B D E C /C /D /B /E
X.9 3 -1 . . B E C D /D /E /B /C
X.10 3 -1 . . /B /E /C /D D E B C
X.11 3 -1 . . /B /C /D /E E C B D
X.12 3 -1 . . /B /D /E /C C D B E
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
= (-1-Sqrt(-7))/2 = -1-b7
C = -E(7)^3-E(7)^5+E(7)^6
D = E(7)^3-E(7)^5-E(7)^6
E = -E(7)^3+E(7)^5-E(7)^6
|