Group action invariants
Degree $n$: | $30$ | |
Transitive number $t$: | $32$ | |
Group: | $S_3\times F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $6$ | |
Generators: | (1,22,12)(2,21,11)(3,5,20,28,24,26,10,18,14,16,29,7)(4,6,19,27,23,25,9,17,13,15,30,8), (1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30) |
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: $S_3$
Degree 10: $F_{5}\times C_2$
Degree 15: $F_5 \times S_3$
Low degree siblings
15T11, 30T23, 30T24Siblings 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 $ | $5$ | $2$ | $( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(13,19)(14,20)(15,27)(16,28)(23,30) (24,29)$ |
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 3,16,10,28)( 4,15, 9,27)( 5,29,18,24)( 6,30,17,23)( 7,14,26,20)( 8,13,25,19)$ |
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 3,28,10,16)( 4,27, 9,15)( 5,24,18,29)( 6,23,17,30)( 7,20,26,14)( 8,19,25,13)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3, 6,10,17)( 4, 5, 9,18)( 7,13,26,19)( 8,14,25,20)(11,22)(12,21) (15,29,27,24)(16,30,28,23)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3,17,10, 6)( 4,18, 9, 5)( 7,19,26,13)( 8,20,25,14)(11,22)(12,21) (15,24,27,29)(16,23,28,30)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,23)( 4,24)( 5,15)( 6,16)( 7, 8)( 9,29)(10,30)(11,22)(12,21)(13,14) (17,28)(18,27)(19,20)(25,26)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,30)( 4,29)( 5,27)( 6,28)( 7,25)( 8,26)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1, 3, 5, 7,10,12,14,16,18,20,22,24,26,28,29)( 2, 4, 6, 8, 9,11,13,15,17,19, 21,23,25,27,30)$ |
$ 6, 6, 6, 6, 3, 3 $ | $10$ | $6$ | $( 1, 3,12,14,22,24)( 2, 4,11,13,21,23)( 5,20,16,29,26,10)( 6,19,15,30,25, 9) ( 7,28,18)( 8,27,17)$ |
$ 12, 12, 3, 3 $ | $10$ | $12$ | $( 1, 3,18,26,22,24, 7,16,12,14,28, 5)( 2, 4,17,25,21,23, 8,15,11,13,27, 6) ( 9,30,19)(10,29,20)$ |
$ 12, 12, 3, 3 $ | $10$ | $12$ | $( 1, 3,29, 7,22,24,20,28,12,14,10,18)( 2, 4,30, 8,21,23,19,27,11,13, 9,17) ( 5,26,16)( 6,25,15)$ |
$ 10, 10, 10 $ | $12$ | $10$ | $( 1, 4,26,27,20,21,14,15, 7, 9)( 2, 3,25,28,19,22,13,16, 8,10)( 5,17,29,11,24, 6,18,30,12,23)$ |
$ 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,14,20,26)( 2, 8,13,19,25)( 3,10,16,22,28)( 4, 9,15,21,27) ( 5,12,18,24,29)( 6,11,17,23,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,12,22)( 2,11,21)( 3,14,24)( 4,13,23)( 5,16,26)( 6,15,25)( 7,18,28) ( 8,17,27)( 9,19,30)(10,20,29)$ |
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 2 2 1 1 2 3 1 1 1 1 . . . . 1 1 1 1 . 1 1 5 1 . . . . . 1 . 1 . . . 1 1 1 1a 2a 4a 4b 4c 4d 2b 2c 15a 6a 12a 12b 10a 5a 3a 2P 1a 1a 2a 2a 2a 2a 1a 1a 15a 3a 6a 6a 5a 5a 3a 3P 1a 2a 4b 4a 4d 4c 2b 2c 5a 2a 4b 4a 10a 5a 1a 5P 1a 2a 4a 4b 4c 4d 2b 2c 3a 6a 12a 12b 2b 1a 3a 7P 1a 2a 4b 4a 4d 4c 2b 2c 15a 6a 12b 12a 10a 5a 3a 11P 1a 2a 4b 4a 4d 4c 2b 2c 15a 6a 12b 12a 10a 5a 3a 13P 1a 2a 4a 4b 4c 4d 2b 2c 15a 6a 12a 12b 10a 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 A -A A -A 1 -1 1 -1 A -A 1 1 1 X.6 1 -1 -A A -A A 1 -1 1 -1 -A A 1 1 1 X.7 1 -1 A -A -A A -1 1 1 -1 A -A -1 1 1 X.8 1 -1 -A A A -A -1 1 1 -1 -A A -1 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 |