Group action invariants
| Degree $n$: | $21$ | |
| Transitive number $t$: | $15$ | |
| Group: | $S_3\times F_7$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,21), (1,13,16)(2,15,17,3,14,18)(4,19,7)(5,21,8,6,20,9)(11,12) |
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 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ $42$: $F_7$ $84$: $F_7 \times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: $F_7$
Low degree siblings
42T43, 42T44, 42T45, 42T52Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $7$ | $3$ | $( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,16)(11,20,17)(12,21,18)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $7$ | $6$ | $( 4,10, 7,19,13,16)( 5,11, 8,20,14,17)( 6,12, 9,21,15,18)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $7$ | $3$ | $( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,16,19)(11,17,20)(12,18,21)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $7$ | $6$ | $( 4,16,13,19, 7,10)( 5,17,14,20, 8,11)( 6,18,15,21, 9,12)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $7$ | $2$ | $( 4,19)( 5,20)( 6,21)( 7,16)( 8,17)( 9,18)(10,13)(11,14)(12,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4, 7,13)( 5, 9,14, 6, 8,15)(10,19,16)(11,21,17,12,20,18)$ |
| $ 6, 6, 6, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4,10, 7,19,13,16)( 5,12, 8,21,14,18)( 6,11, 9,20,15,17)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4,13, 7)( 5,15, 8, 6,14, 9)(10,16,19)(11,18,20,12,17,21)$ |
| $ 6, 6, 6, 2, 1 $ | $21$ | $6$ | $( 2, 3)( 4,16,13,19, 7,10)( 5,18,14,21, 8,12)( 6,17,15,20, 9,11)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $21$ | $2$ | $( 2, 3)( 4,19)( 5,21)( 6,20)( 7,16)( 8,18)( 9,17)(10,13)(11,15)(12,14)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 2, 3)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,20,18)(11,21,16)(12,19,17)$ |
| $ 6, 6, 6, 3 $ | $14$ | $6$ | $( 1, 2, 3)( 4,11, 9,19,14,18)( 5,12, 7,20,15,16)( 6,10, 8,21,13,17)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1, 2, 3)( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,17,21)(11,18,19)(12,16,20)$ |
| $ 6, 6, 6, 3 $ | $14$ | $6$ | $( 1, 2, 3)( 4,17,15,19, 8,12)( 5,18,13,20, 9,10)( 6,16,14,21, 7,11)$ |
| $ 6, 6, 6, 3 $ | $14$ | $6$ | $( 1, 2, 3)( 4,20, 6,19, 5,21)( 7,17, 9,16, 8,18)(10,14,12,13,11,15)$ |
| $ 7, 7, 7 $ | $6$ | $7$ | $( 1, 4, 7,10,13,16,19)( 2, 5, 8,11,14,17,20)( 3, 6, 9,12,15,18,21)$ |
| $ 14, 7 $ | $18$ | $14$ | $( 1, 4, 7,10,13,16,19)( 2, 6, 8,12,14,18,20, 3, 5, 9,11,15,17,21)$ |
| $ 21 $ | $12$ | $21$ | $( 1, 5, 9,10,14,18,19, 2, 6, 7,11,15,16,20, 3, 4, 8,12,13,17,21)$ |
Group invariants
| Order: | $252=2^{2} \cdot 3^{2} \cdot 7$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [252, 26] |
| Character table: not available. |