Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $10$ | |
| Group : | $C_2\times F_7$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,5,4)(2,6,3)(7,19,13,37,26,32)(8,20,14,38,25,31)(9,23,17,40,29,33)(10,24,18,39,30,34)(11,22,15,42,27,36)(12,21,16,41,28,35), (1,18,9,35,20,28)(2,17,10,36,19,27)(3,15,12,31,24,25)(4,16,11,32,23,26)(5,13,8,34,22,30)(6,14,7,33,21,29)(37,42,39,38,41,40) | |
| $|\Aut(F/K)|$: | $6$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 42: $F_7$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 7: $F_7$
Degree 14: $F_7 \times C_2$
Degree 21: 21T4
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, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,37)( 8,38)( 9,40)(10,39)(11,42)(12,41)(13,32)(14,31)(15,36)(16,35)(17,33) (18,34)(19,26)(20,25)(21,28)(22,27)(23,29)(24,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,38)( 8,37)( 9,39)(10,40)(11,41)(12,42)(13,31)(14,32) (15,35)(16,36)(17,34)(18,33)(19,25)(20,26)(21,27)(22,28)(23,30)(24,29)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 3, 5, 2, 4, 6)( 7,14,26, 8,13,25)( 9,18,29,10,17,30)(11,16,27,12,15,28) (19,38,32,20,37,31)(21,42,35,22,41,36)(23,39,33,24,40,34)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 3, 5, 2, 4, 6)( 7,31,26,38,13,20)( 8,32,25,37,14,19)( 9,34,29,39,17,24) (10,33,30,40,18,23)(11,35,27,41,15,21)(12,36,28,42,16,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 4, 5)( 2, 3, 6)( 7,13,26)( 8,14,25)( 9,17,29)(10,18,30)(11,15,27) (12,16,28)(19,37,32)(20,38,31)(21,41,35)(22,42,36)(23,40,33)(24,39,34)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $7$ | $6$ | $( 1, 4, 5)( 2, 3, 6)( 7,32,26,37,13,19)( 8,31,25,38,14,20)( 9,33,29,40,17,23) (10,34,30,39,18,24)(11,36,27,42,15,22)(12,35,28,41,16,21)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $7$ | $6$ | $( 1, 5, 4)( 2, 6, 3)( 7,19,13,37,26,32)( 8,20,14,38,25,31)( 9,23,17,40,29,33) (10,24,18,39,30,34)(11,22,15,42,27,36)(12,21,16,41,28,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 5, 4)( 2, 6, 3)( 7,26,13)( 8,25,14)( 9,29,17)(10,30,18)(11,27,15) (12,28,16)(19,32,37)(20,31,38)(21,35,41)(22,36,42)(23,33,40)(24,34,39)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 6, 4, 2, 5, 3)( 7,20,13,38,26,31)( 8,19,14,37,25,32)( 9,24,17,39,29,34) (10,23,18,40,30,33)(11,21,15,41,27,35)(12,22,16,42,28,36)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $7$ | $6$ | $( 1, 6, 4, 2, 5, 3)( 7,25,13, 8,26,14)( 9,30,17,10,29,18)(11,28,15,12,27,16) (19,31,37,20,32,38)(21,36,41,22,35,42)(23,34,40,24,33,39)$ |
| $ 14, 14, 14 $ | $6$ | $14$ | $( 1, 7,15,24,29,35,38, 2, 8,16,23,30,36,37)( 3, 9,13,22,28,31,39, 4,10,14,21, 27,32,40)( 5,12,17,19,25,34,42, 6,11,18,20,26,33,41)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 8,15,23,29,36,38)( 2, 7,16,24,30,35,37)( 3,10,13,21,28,32,39) ( 4, 9,14,22,27,31,40)( 5,11,17,20,25,33,42)( 6,12,18,19,26,34,41)$ |
Group invariants
| Order: | $84=2^{2} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [84, 7] |
| Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 . .
7 1 . 1 . . . . . . . . . 1 1
1a 2a 2b 2c 6a 6b 3a 6c 6d 3b 6e 6f 14a 7a
2P 1a 1a 1a 1a 3b 3b 3b 3b 3a 3a 3a 3a 7a 7a
3P 1a 2a 2b 2c 2b 2c 1a 2a 2a 1a 2c 2b 14a 7a
5P 1a 2a 2b 2c 6f 6e 3b 6d 6c 3a 6b 6a 14a 7a
7P 1a 2a 2b 2c 6a 6b 3a 6c 6d 3b 6e 6f 2b 1a
11P 1a 2a 2b 2c 6f 6e 3b 6d 6c 3a 6b 6a 14a 7a
13P 1a 2a 2b 2c 6a 6b 3a 6c 6d 3b 6e 6f 14a 7a
X.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
X.3 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 1
X.4 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1
X.5 1 -1 -1 1 A -A -A A /A -/A -/A /A -1 1
X.6 1 -1 -1 1 /A -/A -/A /A A -A -A A -1 1
X.7 1 -1 1 -1 -/A /A -/A /A A -A A -A 1 1
X.8 1 -1 1 -1 -A A -A A /A -/A /A -/A 1 1
X.9 1 1 -1 -1 A A -A -A -/A -/A /A /A -1 1
X.10 1 1 -1 -1 /A /A -/A -/A -A -A A A -1 1
X.11 1 1 1 1 -/A -/A -/A -/A -A -A -A -A 1 1
X.12 1 1 1 1 -A -A -A -A -/A -/A -/A -/A 1 1
X.13 6 . -6 . . . . . . . . . 1 -1
X.14 6 . 6 . . . . . . . . . -1 -1
A = -E(3)
= (1-Sqrt(-3))/2 = -b3
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