Group action invariants
Degree $n$: | $30$ | |
Transitive number $t$: | $50$ | |
Group: | $F_{16}$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22), (1,18,21,25,15,3,12,10,19,7,6,30,27,13,23)(2,17,22,26,16,4,11,9,20,8,5,29,28,14,24) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $5$: $C_5$ $15$: $C_{15}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 5: $C_5$
Degree 6: None
Degree 10: None
Degree 15: $C_{15}$
Low degree siblings
16T447, 20T67Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 3, 5)( 2, 4, 6)( 7,23,15)( 8,24,16)( 9,28,21)(10,27,22)(11,30,17) (12,29,18)(13,26,20)(14,25,19)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 5, 3)( 2, 6, 4)( 7,15,23)( 8,16,24)( 9,21,28)(10,22,27)(11,17,30) (12,18,29)(13,20,26)(14,19,25)$ |
$ 5, 5, 5, 5, 5, 5 $ | $16$ | $5$ | $( 1, 7,25,27,12)( 2, 8,26,28,11)( 3,23,19,21,30)( 4,24,20,22,29) ( 5,16,14, 9,17)( 6,15,13,10,18)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1, 9,24,11,13, 3,27,15,29,25, 5,21, 7,17,20)( 2,10,23,12,14, 4,28,16,30,26, 6,22, 8,18,19)$ |
$ 5, 5, 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,11,27,25, 7)( 2,12,28,26, 8)( 3,29,21,20,24)( 4,30,22,19,23) ( 5,17, 9,13,15)( 6,18,10,14,16)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,13,29, 7, 9, 3,25,17,24,27, 5,20,11,15,21)( 2,14,30, 8,10, 4,26,18,23,28, 6,19,12,16,22)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,15,19,27,18, 3, 7,13,21,12, 6,23,25,10,30)( 2,16,20,28,17, 4, 8,14,22,11, 5,24,26, 9,29)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,17,21,25,15, 3,11, 9,20, 7, 5,29,27,13,24)( 2,18,22,26,16, 4,12,10,19, 8, 6,30,28,14,23)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,19,18, 7,21, 6,25,30,15,27, 3,13,12,23,10)( 2,20,17, 8,22, 5,26,29,16,28, 4,14,11,24, 9)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,21,15,11,20, 5,27,24,17,25, 3, 9, 7,29,13)( 2,22,16,12,19, 6,28,23,18,26, 4,10, 8,30,14)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,23,13,27,30, 6, 7,19,10,12, 3,15,25,21,18)( 2,24,14,28,29, 5, 8,20, 9,11, 4,16,26,22,17)$ |
$ 5, 5, 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,25,11, 8,28)( 2,26,12, 7,27)( 3,19,29,23,22)( 4,20,30,24,21) ( 5,14,17,15, 9)( 6,13,18,16,10)$ |
$ 5, 5, 5, 5, 5, 5 $ | $16$ | $5$ | $( 1,27, 7,11,25)( 2,28, 8,12,26)( 3,21,24,29,20)( 4,22,23,30,19) ( 5, 9,15,17,13)( 6,10,16,18,14)$ |
$ 15, 15 $ | $16$ | $15$ | $( 1,29, 9,25,24, 5,11,21,13, 7, 3,17,27,20,15)( 2,30,10,26,23, 6,12,22,14, 8, 4,18,28,19,16)$ |
Group invariants
Order: | $240=2^{4} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [240, 191] |
Character table: |
2 4 4 . . . . . . . . . . . . . . 3 1 . 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 1 1 1 1 1 1a 2a 3a 3b 5a 15a 5b 15b 15c 15d 15e 15f 15g 5c 5d 15h 2P 1a 1a 3b 3a 5c 15g 5d 15h 15e 15f 15d 15c 15b 5b 5a 15a 3P 1a 2a 1a 1a 5d 5b 5c 5a 5d 5c 5a 5b 5d 5a 5b 5c 5P 1a 2a 3b 3a 1a 3a 1a 3a 3a 3a 3b 3b 3b 1a 1a 3b 7P 1a 2a 3a 3b 5c 15c 5d 15d 15b 15a 15h 15g 15e 5b 5a 15f 11P 1a 2a 3b 3a 5a 15f 5b 15e 15g 15h 15b 15a 15c 5c 5d 15d 13P 1a 2a 3a 3b 5d 15d 5c 15c 15a 15b 15g 15h 15f 5a 5b 15e X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 B /C /B C B /B C /C B C /C /B X.3 1 1 1 1 C B /C /B C /C /B B C /B B /C X.4 1 1 1 1 /C /B C B /C C B /B /C B /B C X.5 1 1 1 1 /B C B /C /B B /C C /B /C C B X.6 1 1 A /A 1 /A 1 /A /A /A A A A 1 1 A X.7 1 1 /A A 1 A 1 A A A /A /A /A 1 1 /A X.8 1 1 A /A B D /B G E F /D /G /F C /C /E X.9 1 1 A /A C E /C F G D /E /F /D /B B /G X.10 1 1 A /A /C F C E D G /F /E /G B /B /D X.11 1 1 A /A /B G B D F E /G /D /E /C C /F X.12 1 1 /A A B /G /B /D /F /E G D E C /C F X.13 1 1 /A A C /F /C /E /D /G F E G /B B D X.14 1 1 /A A /C /E C /F /G /D E F D B /B G X.15 1 1 /A A /B /D B /G /E /F D G F /C C E X.16 15 -1 . . . . . . . . . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(5)^4 C = E(5)^3 D = E(15)^11 E = E(15)^2 F = E(15)^8 G = E(15)^14 |