Group action invariants
| Degree $n$ : | $30$ | |
| Transitive number $t$ : | $21$ | |
| Group : | $C_2\times S_3\times D_5$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23), (1,6)(2,5)(3,4)(7,30)(8,29)(9,27)(10,28)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19), (1,11)(2,12)(5,25)(6,26)(7,17)(8,18)(13,23)(14,24)(19,29)(20,30) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 7 4: $C_2^2$ x 7 6: $S_3$ 8: $C_2^3$ 10: $D_{5}$ 12: $D_{6}$ x 3 20: $D_{10}$ x 3 24: $S_3 \times C_2^2$ 40: 20T8 60: $D_5\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: $D_{5}$
Degree 6: $D_{6}$
Degree 10: $D_{10}$
Degree 15: $D_5\times S_3$
Low degree siblings
30T21 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3, 9)( 4,10)( 5,17)( 6,18)( 7,25)( 8,26)(13,20)(14,19)(15,27)(16,28)(23,30) (24,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3,23)( 4,24)( 5,15)( 6,16)( 9,30)(10,29)(11,21)(12,22)(17,27)(18,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $15$ | $2$ | $( 3,30)( 4,29)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22)(13,20) (14,19)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(11,12)(13,19)(14,20)(15,28) (16,27)(21,22)(23,29)(24,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,24)( 4,23)( 5,16)( 6,15)( 7, 8)( 9,29)(10,30)(11,22)(12,21)(13,14) (17,28)(18,27)(19,20)(25,26)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,29)( 4,30)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,19)(14,20)(15,18)(16,17)$ |
| $ 30 $ | $4$ | $30$ | $( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19, 22,24,26,27,30)$ |
| $ 6, 6, 6, 6, 6 $ | $10$ | $6$ | $( 1, 3,11,13,21,23)( 2, 4,12,14,22,24)( 5,20,15,30,25, 9)( 6,19,16,29,26,10) ( 7,28,17, 8,27,18)$ |
| $ 10, 10, 10 $ | $6$ | $10$ | $( 1, 3,25,28,19,22,14,16, 7, 9)( 2, 4,26,27,20,21,13,15, 8,10)( 5,18,29,12,24, 6,17,30,11,23)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1, 4, 5, 7,10,11,14,15,17,19,21,24,25,27,29)( 2, 3, 6, 8, 9,12,13,16,18,20, 22,23,26,28,30)$ |
| $ 6, 6, 6, 6, 3, 3 $ | $10$ | $6$ | $( 1, 4,11,14,21,24)( 2, 3,12,13,22,23)( 5,19,15,29,25,10)( 6,20,16,30,26, 9) ( 7,27,17)( 8,28,18)$ |
| $ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 4,25,27,19,21,14,15, 7,10)( 2, 3,26,28,20,22,13,16, 8, 9)( 5,17,29,11,24) ( 6,18,30,12,23)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1, 5,10,14,17,21,25,29, 4, 7,11,15,19,24,27)( 2, 6, 9,13,18,22,26,30, 3, 8, 12,16,20,23,28)$ |
| $ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 5,19,24, 7,11,25,29,14,17)( 2, 6,20,23, 8,12,26,30,13,18)( 3,28,22,16, 9) ( 4,27,21,15,10)$ |
| $ 30 $ | $4$ | $30$ | $( 1, 6,10,13,17,22,25,30, 4, 8,11,16,19,23,27, 2, 5, 9,14,18,21,26,29, 3, 7, 12,15,20,24,28)$ |
| $ 10, 10, 10 $ | $6$ | $10$ | $( 1, 6,19,23, 7,12,25,30,14,18)( 2, 5,20,24, 8,11,26,29,13,17)( 3,27,22,15, 9, 4,28,21,16,10)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$ |
| $ 10, 10, 10 $ | $2$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,11,21)( 2,12,22)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,20,30)(10,19,29)$ |
| $ 6, 6, 6, 6, 6 $ | $2$ | $6$ | $( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$ |
| $ 10, 10, 10 $ | $2$ | $10$ | $( 1,13,25, 8,19, 2,14,26, 7,20)( 3,15,28,10,22, 4,16,27, 9,21)( 5,18,29,12,24, 6,17,30,11,23)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,14,25, 7,19)( 2,13,26, 8,20)( 3,16,28, 9,22)( 4,15,27,10,21) ( 5,17,29,11,24)( 6,18,30,12,23)$ |
Group invariants
| Order: | $120=2^{3} \cdot 3 \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [120, 42] |
| Character table: Data not available. |