Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $30$ | |
| Group : | $D_7:A_4$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,23,37,15,36,8,29,2,24,38,16,35,7,30)(3,22,39,13,31,10,27)(4,21,40,14,32,9,28)(5,20,41,17,33,11,25,6,19,42,18,34,12,26), (1,22,26,15,40,33)(2,21,25,16,39,34)(3,20,30,14,41,36)(4,19,29,13,42,35)(5,24,27,17,38,32)(6,23,28,18,37,31)(7,10,11,8,9,12) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 6: $C_6$ 12: $A_4$ 24: $A_4\times C_2$ 42: $F_7$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 7: $F_7$
Degree 14: None
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, 2, 1, 1, 1, 1 $ | $21$ | $2$ | $( 5, 6)( 7,37)( 8,38)( 9,40)(10,39)(11,41)(12,42)(13,31)(14,32)(15,35)(16,36) (17,33)(18,34)(19,26)(20,25)(21,28)(22,27)(23,30)(24,29)$ |
| $ 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 $ | $3$ | $2$ | $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)(25,26)(27,28)(31,32) (33,34)(39,40)(41,42)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,38)( 8,37)( 9,39)(10,40)(11,41)(12,42)(13,32)(14,31) (15,36)(16,35)(17,33)(18,34)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,13,25)( 8,14,26)( 9,17,30)(10,18,29)(11,15,28) (12,16,27)(19,37,31)(20,38,32)(21,42,35)(22,41,36)(23,40,34)(24,39,33)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $28$ | $6$ | $( 1, 3, 5, 2, 4, 6)( 7,31,25,38,14,20)( 8,32,26,37,13,19)( 9,34,29,39,18,23) (10,33,30,40,17,24)(11,36,27,41,15,21)(12,35,28,42,16,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,13)( 8,26,14)( 9,30,17)(10,29,18)(11,28,15) (12,27,16)(19,31,37)(20,32,38)(21,35,42)(22,36,41)(23,34,40)(24,33,39)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $28$ | $6$ | $( 1, 5, 4, 2, 6, 3)( 7,19,14,38,26,31)( 8,20,13,37,25,32)( 9,23,17,39,29,33) (10,24,18,40,30,34)(11,22,16,41,28,35)(12,21,15,42,27,36)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 7,16,24,29,36,37)( 2, 8,15,23,30,35,38)( 3, 9,13,21,27,32,39, 4,10,14,22, 28,31,40)( 5,11,18,20,25,34,41, 6,12,17,19,26,33,42)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 7,16,24,29,36,37)( 2, 8,15,23,30,35,38)( 3,10,13,22,27,31,39) ( 4, 9,14,21,28,32,40)( 5,12,18,19,25,33,41)( 6,11,17,20,26,34,42)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,23,29,35,37, 2, 7,15,24,30,36,38)( 3, 9,13,21,27,32,39, 4,10,14,22, 28,31,40)( 5,12,18,19,25,33,41)( 6,11,17,20,26,34,42)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,23,29,35,37, 2, 7,15,24,30,36,38)( 3,10,13,22,27,31,39) ( 4, 9,14,21,28,32,40)( 5,11,18,20,25,34,41, 6,12,17,19,26,33,42)$ |
Group invariants
| Order: | $168=2^{3} \cdot 3 \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [168, 49] |
| Character table: |
2 3 3 3 3 1 1 1 1 2 2 2 2
3 1 . . 1 1 1 1 1 . . . .
7 1 . 1 . . . . . 1 1 1 1
1a 2a 2b 2c 3a 6a 3b 6b 14a 7a 14b 14c
2P 1a 1a 1a 1a 3b 3b 3a 3a 7a 7a 7a 7a
3P 1a 2a 2b 2c 1a 2c 1a 2c 14c 7a 14a 14b
5P 1a 2a 2b 2c 3b 6b 3a 6a 14b 7a 14c 14a
7P 1a 2a 2b 2c 3a 6a 3b 6b 2b 1a 2b 2b
11P 1a 2a 2b 2c 3b 6b 3a 6a 14c 7a 14a 14b
13P 1a 2a 2b 2c 3a 6a 3b 6b 14a 7a 14b 14c
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 -1 1 -1 A -A /A -/A 1 1 1 1
X.4 1 -1 1 -1 /A -/A A -A 1 1 1 1
X.5 1 1 1 1 A A /A /A 1 1 1 1
X.6 1 1 1 1 /A /A A A 1 1 1 1
X.7 3 -1 -1 3 . . . . -1 3 -1 -1
X.8 3 1 -1 -3 . . . . -1 3 -1 -1
X.9 6 . 6 . . . . . -1 -1 -1 -1
X.10 6 . -2 . . . . . B -1 D C
X.11 6 . -2 . . . . . C -1 B D
X.12 6 . -2 . . . . . D -1 C B
A = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
B = E(7)-E(7)^2-E(7)^3-E(7)^4-E(7)^5+E(7)^6
C = -E(7)-E(7)^2+E(7)^3+E(7)^4-E(7)^5-E(7)^6
D = -E(7)+E(7)^2-E(7)^3-E(7)^4+E(7)^5-E(7)^6
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