Group action invariants
| Degree $n$ : | $42$ | |
| Transitive number $t$ : | $43$ | |
| Group : | $S_3\times F_7$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,21,6,20,3,24)(2,22,5,19,4,23)(7,15,12,13,9,18)(8,16,11,14,10,17)(25,40,29,37,28,41)(26,39,30,38,27,42)(31,33,35,32,34,36), (1,2)(3,5)(4,6)(7,26,14,8,25,13)(9,30,16,11,28,18)(10,29,15,12,27,17)(19,32,38,20,31,37)(21,35,40,23,33,42)(22,36,39,24,34,41) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 12: $D_{6}$ 18: $S_3\times C_3$ 42: $F_7$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 7: $F_7$
Degree 14: $F_7 \times C_2$
Degree 21: 21T15
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 $ | $7$ | $3$ | $( 7,14,25)( 8,13,26)( 9,16,28)(10,15,27)(11,18,30)(12,17,29)(19,38,31) (20,37,32)(21,40,33)(22,39,34)(23,42,35)(24,41,36)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $7$ | $3$ | $( 7,25,14)( 8,26,13)( 9,28,16)(10,27,15)(11,30,18)(12,29,17)(19,31,38) (20,32,37)(21,33,40)(22,34,39)(23,35,42)(24,36,41)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 6)( 4, 5)( 7,19,14,38,25,31)( 8,20,13,37,26,32)( 9,23,16,42,28,35) (10,24,15,41,27,36)(11,21,18,40,30,33)(12,22,17,39,29,34)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 1, 1 $ | $21$ | $6$ | $( 3, 6)( 4, 5)( 7,31,25,38,14,19)( 8,32,26,37,13,20)( 9,35,28,42,16,23) (10,36,27,41,15,24)(11,33,30,40,18,21)(12,34,29,39,17,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $21$ | $2$ | $( 3, 6)( 4, 5)( 7,38)( 8,37)( 9,42)(10,41)(11,40)(12,39)(13,32)(14,31)(15,36) (16,35)(17,34)(18,33)(19,25)(20,26)(21,30)(22,29)(23,28)(24,27)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,20,14,37,25,32)( 8,19,13,38,26,31)( 9,21,16,40,28,33) (10,22,15,39,27,34)(11,23,18,42,30,35)(12,24,17,41,29,36)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $7$ | $6$ | $( 1, 2)( 3, 4)( 5, 6)( 7,32,25,37,14,20)( 8,31,26,38,13,19)( 9,33,28,40,16,21) (10,34,27,39,15,22)(11,35,30,42,18,23)(12,36,29,41,17,24)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $7$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7,37)( 8,38)( 9,40)(10,39)(11,42)(12,41)(13,31)(14,32) (15,34)(16,33)(17,36)(18,35)(19,26)(20,25)(21,28)(22,27)(23,30)(24,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,23) (22,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,36)(37,38)(39,41)(40,42)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 5)( 4, 6)( 7,13,25, 8,14,26)( 9,18,28,11,16,30)(10,17,27,12,15,29) (19,37,31,20,38,32)(21,42,33,23,40,35)(22,41,34,24,39,36)$ |
| $ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ | $21$ | $6$ | $( 1, 2)( 3, 5)( 4, 6)( 7,26,14, 8,25,13)( 9,30,16,11,28,18)(10,29,15,12,27,17) (19,32,38,20,31,37)(21,35,40,23,33,42)(22,36,39,24,34,41)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7, 9,12)( 8,10,11)(13,15,18)(14,16,17)(19,22,23) (20,21,24)(25,28,29)(26,27,30)(31,34,35)(32,33,36)(37,40,41)(38,39,42)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,16,29)( 8,15,30)( 9,17,25)(10,18,26)(11,13,27) (12,14,28)(19,39,35)(20,40,36)(21,41,32)(22,42,31)(23,38,34)(24,37,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 3, 6)( 2, 4, 5)( 7,28,17)( 8,27,18)( 9,29,14)(10,30,13)(11,26,15) (12,25,16)(19,34,42)(20,33,41)(21,36,37)(22,35,38)(23,31,39)(24,32,40)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,21,17,37,28,36)( 8,22,18,38,27,35)( 9,24,14,40,29,32) (10,23,13,39,30,31)(11,19,15,42,26,34)(12,20,16,41,25,33)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,33,29,37,16,24)( 8,34,30,38,15,23)( 9,36,25,40,17,20) (10,35,26,39,18,19)(11,31,27,42,13,22)(12,32,28,41,14,21)$ |
| $ 6, 6, 6, 6, 6, 6, 6 $ | $14$ | $6$ | $( 1, 4, 6, 2, 3, 5)( 7,40,12,37, 9,41)( 8,39,11,38,10,42)(13,34,18,31,15,35) (14,33,17,32,16,36)(19,27,23,26,22,30)(20,28,24,25,21,29)$ |
| $ 7, 7, 7, 7, 7, 7 $ | $6$ | $7$ | $( 1, 7,14,19,25,31,38)( 2, 8,13,20,26,32,37)( 3, 9,16,22,28,34,39) ( 4,10,15,21,27,33,40)( 5,11,18,24,30,36,41)( 6,12,17,23,29,35,42)$ |
| $ 14, 14, 14 $ | $18$ | $14$ | $( 1, 8,14,20,25,32,38, 2, 7,13,19,26,31,37)( 3,11,16,24,28,36,39, 5, 9,18,22, 30,34,41)( 4,12,15,23,27,35,40, 6,10,17,21,29,33,42)$ |
| $ 21, 21 $ | $12$ | $21$ | $( 1, 9,17,19,28,35,38, 3,12,14,22,29,31,39, 6, 7,16,23,25,34,42) ( 2,10,18,20,27,36,37, 4,11,13,21,30,32,40, 5, 8,15,24,26,33,41)$ |
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [252, 26] |
| Character table: Data not available. |