Group action invariants
Degree $n$: | $30$ | |
Transitive number $t$: | $17$ | |
Group: | $C_2\times C_3:F_5$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (3,6,9,18)(4,5,10,17)(7,14,25,19)(8,13,26,20)(11,21)(12,22)(15,29,27,24)(16,30,28,23), (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) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$, $C_3 : C_4$ x 2 $20$: $F_5$ $24$: 24T6 $40$: $F_{5}\times C_2$ $60$: $C_{15} : C_4$ 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: $F_{5}\times C_2$
Degree 15: $C_{15} : C_4$
Low degree siblings
30T17Siblings 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 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1 $ | $15$ | $4$ | $( 3, 6, 9,18)( 4, 5,10,17)( 7,14,25,19)( 8,13,26,20)(11,21)(12,22) (15,29,27,24)(16,30,28,23)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3, 9)( 4,10)( 5,17)( 6,18)( 7,25)( 8,26)(13,20)(14,19)(15,27)(16,28)(23,30) (24,29)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 1, 1 $ | $15$ | $4$ | $( 3,18, 9, 6)( 4,17,10, 5)( 7,19,25,14)( 8,20,26,13)(11,21)(12,22) (15,24,27,29)(16,23,28,30)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3, 5, 9,17)( 4, 6,10,18)( 7,13,25,20)( 8,14,26,19)(11,22)(12,21) (15,30,27,23)(16,29,28,24)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(11,12)(13,19)(14,20)(15,28) (16,27)(21,22)(23,29)(24,30)$ |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $15$ | $4$ | $( 1, 2)( 3,17, 9, 5)( 4,18,10, 6)( 7,20,25,13)( 8,19,26,14)(11,22)(12,21) (15,23,27,30)(16,24,28,29)$ |
$ 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 $ | $10$ | $6$ | $( 1, 3,11,13,21,23)( 2, 4,12,14,22,24)( 5,20,15,30,25, 9)( 6,19,16,29,26,10) ( 7,28,17, 8,27,18)$ |
$ 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 $ | $10$ | $6$ | $( 1, 4,11,14,21,24)( 2, 3,12,13,22,23)( 5,19,15,29,25,10)( 6,20,16,30,26, 9) ( 7,27,17)( 8,28,18)$ |
$ 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$ |
$ 10, 10, 10 $ | $4$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$ |
$ 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 $ | $2$ | $6$ | $( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$ |
$ 15, 15 $ | $4$ | $15$ | $( 1,15,29,14,27,11,25,10,24, 7,21, 5,19, 4,17)( 2,16,30,13,28,12,26, 9,23, 8, 22, 6,20, 3,18)$ |
$ 30 $ | $4$ | $30$ | $( 1,16,29,13,27,12,25, 9,24, 8,21, 6,19, 3,17, 2,15,30,14,28,11,26,10,23, 7, 22, 5,20, 4,18)$ |
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [120, 41] |
Character table: |
2 3 3 3 3 3 3 3 3 1 2 1 2 1 1 2 2 1 1 3 1 . 1 . 1 . 1 . 1 1 1 1 1 1 1 1 1 1 5 1 . . . 1 . . . 1 . 1 . 1 1 1 1 1 1 1a 4a 2a 4b 2b 4c 2c 4d 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b 2P 1a 2a 1a 2a 1a 2a 1a 2a 15a 3a 15a 3a 5a 5a 3a 3a 15b 15b 3P 1a 4b 2a 4a 2b 4d 2c 4c 10a 2c 5a 2a 5a 10a 1a 2b 5a 10a 5P 1a 4a 2a 4b 2b 4c 2c 4d 6c 6a 3a 6b 1a 2b 3a 6c 3a 6c 7P 1a 4b 2a 4a 2b 4d 2c 4c 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a 11P 1a 4b 2a 4a 2b 4d 2c 4c 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a 13P 1a 4a 2a 4b 2b 4c 2c 4d 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a 17P 1a 4a 2a 4b 2b 4c 2c 4d 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b 19P 1a 4b 2a 4a 2b 4d 2c 4c 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b 23P 1a 4b 2a 4a 2b 4d 2c 4c 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b 29P 1a 4a 2a 4b 2b 4c 2c 4d 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a X.1 1 1 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 -1 1 -1 X.3 1 -1 1 -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 -1 1 -1 X.5 1 A -1 -A -1 -A 1 A -1 1 1 -1 1 -1 1 -1 1 -1 X.6 1 -A -1 A -1 A 1 -A -1 1 1 -1 1 -1 1 -1 1 -1 X.7 1 A -1 -A 1 A -1 -A 1 -1 1 -1 1 1 1 1 1 1 X.8 1 -A -1 A 1 -A -1 A 1 -1 1 -1 1 1 1 1 1 1 X.9 2 . -2 . -2 . 2 . 1 -1 -1 1 2 -2 -1 1 -1 1 X.10 2 . -2 . 2 . -2 . -1 1 -1 1 2 2 -1 -1 -1 -1 X.11 2 . 2 . -2 . -2 . 1 1 -1 -1 2 -2 -1 1 -1 1 X.12 2 . 2 . 2 . 2 . -1 -1 -1 -1 2 2 -1 -1 -1 -1 X.13 4 . . . 4 . . . -1 . -1 . -1 -1 4 4 -1 -1 X.14 4 . . . -4 . . . 1 . -1 . -1 1 4 -4 -1 1 X.15 4 . . . -4 . . . B . -B . -1 1 -2 2 -/B /B X.16 4 . . . -4 . . . /B . -/B . -1 1 -2 2 -B B X.17 4 . . . 4 . . . -/B . -/B . -1 -1 -2 -2 -B -B X.18 4 . . . 4 . . . -B . -B . -1 -1 -2 -2 -/B -/B A = -E(4) = -Sqrt(-1) = -i B = -E(15)-E(15)^2-E(15)^4-E(15)^8 = (-1-Sqrt(-15))/2 = -1-b15 |