Group action invariants
Degree $n$: | $30$ | |
Transitive number $t$: | $24$ | |
Group: | $S_3\times F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,7,13,20,26)(2,8,14,19,25)(3,29,15,11,28,24,9,5,22,18)(4,30,16,12,27,23,10,6,21,17), (1,4,18,25,22,23,7,16,11,14,28,6)(2,3,17,26,21,24,8,15,12,13,27,5)(9,30,20,10,29,19) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$ $20$: $F_5$ $24$: $S_3 \times C_4$ $40$: $F_{5}\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: $F_5$
Degree 6: $D_{6}$
Degree 10: $F_5$
Degree 15: $F_5 \times S_3$
Low degree siblings
15T11, 30T23, 30T32Siblings 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 $ | $5$ | $2$ | $( 3, 9)( 4,10)( 5,18)( 6,17)( 7,26)( 8,25)(13,20)(14,19)(15,28)(16,27)(23,30) (24,29)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 3,24)( 4,23)( 5,15)( 6,16)( 9,29)(10,30)(11,22)(12,21)(17,27)(18,28)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $15$ | $2$ | $( 3,29)( 4,30)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(13,20) (14,19)(15,18)(16,17)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3, 6, 9,17)( 4, 5,10,18)( 7,14,26,19)( 8,13,25,20)(11,21)(12,22) (15,30,28,23)(16,29,27,24)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $5$ | $4$ | $( 1, 2)( 3,16, 9,27)( 4,15,10,28)( 5,30,18,23)( 6,29,17,24)( 7,14,26,19) ( 8,13,25,20)(11,12)(21,22)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3,17, 9, 6)( 4,18,10, 5)( 7,19,26,14)( 8,20,25,13)(11,21)(12,22) (15,23,28,30)(16,24,27,29)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $5$ | $4$ | $( 1, 2)( 3,27, 9,16)( 4,28,10,15)( 5,23,18,30)( 6,24,17,29)( 7,19,26,14) ( 8,20,25,13)(11,12)(21,22)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1, 3, 5, 7, 9,11,13,15,18,20,22,24,26,28,29)( 2, 4, 6, 8,10,12,14,16,17,19, 21,23,25,27,30)$ |
$ 6, 6, 6, 6, 3, 3 $ | $10$ | $6$ | $( 1, 3,11,13,22,24)( 2, 4,12,14,21,23)( 5,20,15,29,26, 9)( 6,19,16,30,25,10) ( 7,28,18)( 8,27,17)$ |
$ 10, 10, 5, 5 $ | $12$ | $10$ | $( 1, 3,26,28,20,22,13,15, 7, 9)( 2, 4,25,27,19,21,14,16, 8,10)( 5,18,29,11,24) ( 6,17,30,12,23)$ |
$ 12, 12, 6 $ | $10$ | $12$ | $( 1, 4,18,25,22,23, 7,16,11,14,28, 6)( 2, 3,17,26,21,24, 8,15,12,13,27, 5) ( 9,30,20,10,29,19)$ |
$ 12, 12, 6 $ | $10$ | $12$ | $( 1, 4,29, 8,22,23,20,27,11,14, 9,17)( 2, 3,30, 7,21,24,19,28,12,13,10,18) ( 5,25,15, 6,26,16)$ |
$ 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,13,20,26)( 2, 8,14,19,25)( 3, 9,15,22,28)( 4,10,16,21,27) ( 5,11,18,24,29)( 6,12,17,23,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,11,22)( 2,12,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,28) ( 8,17,27)( 9,20,29)(10,19,30)$ |
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [120, 36] |
Character table: |
2 3 3 3 3 3 3 3 3 . 2 1 2 2 1 2 3 1 1 . . . 1 . 1 1 1 . 1 1 1 1 5 1 . 1 . . . . . 1 . 1 . . 1 1 1a 2a 2b 2c 4a 4b 4c 4d 15a 6a 10a 12a 12b 5a 3a 2P 1a 1a 1a 1a 2a 2a 2a 2a 15a 3a 5a 6a 6a 5a 3a 3P 1a 2a 2b 2c 4c 4d 4a 4b 5a 2a 10a 4d 4b 5a 1a 5P 1a 2a 2b 2c 4a 4b 4c 4d 3a 6a 2b 12a 12b 1a 3a 7P 1a 2a 2b 2c 4c 4d 4a 4b 15a 6a 10a 12b 12a 5a 3a 11P 1a 2a 2b 2c 4c 4d 4a 4b 15a 6a 10a 12b 12a 5a 3a 13P 1a 2a 2b 2c 4a 4b 4c 4d 15a 6a 10a 12a 12b 5a 3a X.1 1 1 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 1 1 1 X.3 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 X.5 1 -1 -1 1 A -A -A A 1 -1 -1 -A A 1 1 X.6 1 -1 -1 1 -A A A -A 1 -1 -1 A -A 1 1 X.7 1 -1 1 -1 A A -A -A 1 -1 1 A -A 1 1 X.8 1 -1 1 -1 -A -A A A 1 -1 1 -A A 1 1 X.9 2 2 . . . -2 . -2 -1 -1 . 1 1 2 -1 X.10 2 2 . . . 2 . 2 -1 -1 . -1 -1 2 -1 X.11 2 -2 . . . B . -B -1 1 . -A A 2 -1 X.12 2 -2 . . . -B . B -1 1 . A -A 2 -1 X.13 4 . -4 . . . . . -1 . 1 . . -1 4 X.14 4 . 4 . . . . . -1 . -1 . . -1 4 X.15 8 . . . . . . . 1 . . . . -2 -4 A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i |