Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $31$ | |
| Group : | $D_7:A_4$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,9,19)(2,10,20)(3,12,23)(4,11,24)(5,7,21)(6,8,22)(13,34,30)(14,33,29)(15,32,26)(16,31,25)(17,35,28)(18,36,27)(37,39,41)(38,40,42), (1,13,33,38,28,11)(2,14,34,37,27,12)(3,18,35,40,26,7)(4,17,36,39,25,8)(5,15,32,42,29,10)(6,16,31,41,30,9)(19,24,22)(20,23,21) | |
| $|\Aut(F/K)|$: | $2$ |
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: $A_4$
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, 1, 1, 1, 1, 1, 1 $ | $7$ | $2$ | $( 7,38)( 8,37)( 9,40)(10,39)(11,41)(12,42)(13,31)(14,32)(15,36)(16,35)(17,34) (18,33)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)$ |
| $ 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, 1, 1 $ | $21$ | $2$ | $( 3, 4)( 5, 6)( 7,38)( 8,37)( 9,39)(10,40)(11,42)(12,41)(13,32)(14,31)(15,36) (16,35)(17,33)(18,34)(19,26)(20,25)(21,27)(22,28)(23,30)(24,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,13,25)( 8,14,26)( 9,17,30)(10,18,29)(11,15,27) (12,16,28)(19,38,31)(20,37,32)(21,42,35)(22,41,36)(23,40,34)(24,39,33)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $28$ | $6$ | $( 1, 3, 5)( 2, 4, 6)( 7,31,25,38,13,19)( 8,32,26,37,14,20)( 9,34,30,40,17,23) (10,33,29,39,18,24)(11,36,27,41,15,22)(12,35,28,42,16,21)$ |
| $ 6, 6, 6, 6, 6, 6, 3, 3 $ | $28$ | $6$ | $( 1, 5, 3)( 2, 6, 4)( 7,19,13,38,25,31)( 8,20,14,37,26,32)( 9,23,17,40,30,34) (10,24,18,39,29,33)(11,22,15,41,27,36)(12,21,16,42,28,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $28$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,25,13)( 8,26,14)( 9,30,17)(10,29,18)(11,27,15) (12,28,16)(19,31,38)(20,32,37)(21,35,42)(22,36,41)(23,34,40)(24,33,39)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 7,16,23,30,35,38)( 2, 8,15,24,29,36,37)( 3, 9,13,21,28,31,40) ( 4,10,14,22,27,32,39)( 5,12,17,19,25,34,42)( 6,11,18,20,26,33,41)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 7,16,23,30,35,38)( 2, 8,15,24,29,36,37)( 3,10,13,22,28,32,40, 4, 9,14,21, 27,31,39)( 5,11,17,20,25,33,42, 6,12,18,19,26,34,41)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,24,30,36,38, 2, 7,15,23,29,35,37)( 3, 9,13,21,28,31,40) ( 4,10,14,22,27,32,39)( 5,11,17,20,25,33,42, 6,12,18,19,26,34,41)$ |
| $ 14, 14, 7, 7 $ | $6$ | $14$ | $( 1, 8,16,24,30,36,38, 2, 7,15,23,29,35,37)( 3,10,13,22,28,32,40, 4, 9,14,21, 27,31,39)( 5,12,17,19,25,34,42)( 6,11,18,20,26,33,41)$ |
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 6b 3b 7a 14a 14b 14c
2P 1a 1a 1a 1a 3b 3b 3a 3a 7a 7a 7a 7a
3P 1a 2a 2b 2c 1a 2a 2a 1a 7a 14b 14c 14a
5P 1a 2a 2b 2c 3b 6b 6a 3a 7a 14c 14a 14b
7P 1a 2a 2b 2c 3a 6a 6b 3b 1a 2b 2b 2b
11P 1a 2a 2b 2c 3b 6b 6a 3a 7a 14b 14c 14a
13P 1a 2a 2b 2c 3a 6a 6b 3b 7a 14a 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 -3 -1 1 . . . . 3 -1 -1 -1
X.8 3 3 -1 -1 . . . . 3 -1 -1 -1
X.9 6 . 6 . . . . . -1 -1 -1 -1
X.10 6 . -2 . . . . . -1 B C D
X.11 6 . -2 . . . . . -1 C D B
X.12 6 . -2 . . . . . -1 D B C
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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