Group action invariants
Degree $n$: | $20$ | |
Transitive number $t$: | $14$ | |
Group: | $C_5\times A_4$ | |
Parity: | $1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $5$ | |
Generators: | (1,8,10,13,20,2,5,12,14,17,4,6,9,16,18)(3,7,11,15,19), (1,12,17,8,13,4,9,20,5,16)(2,11,18,7,14,3,10,19,6,15) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $5$: $C_5$ $12$: $A_4$ $15$: $C_{15}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $A_4$
Degree 5: $C_5$
Degree 10: None
Low degree siblings
30T11Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$ |
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $4$ | $3$ | $( 2, 4, 3)( 6, 8, 7)(10,12,11)(14,16,15)(18,20,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,13, 5,17, 9)( 2,14, 6,18,10)( 3,15, 7,19,11)( 4,16, 8,20,12)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1,13, 5,17, 9)( 2,15, 8,18,11, 4,14, 7,20,10, 3,16, 6,19,12)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1,13, 5,17, 9)( 2,16, 7,18,12, 3,14, 8,19,10, 4,15, 6,20,11)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,14, 5,18, 9, 2,13, 6,17,10)( 3,16, 7,20,11, 4,15, 8,19,12)$ |
$ 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,17,13, 9, 5)( 2,18,14,10, 6)( 3,19,15,11, 7)( 4,20,16,12, 8)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1,17,13, 9, 5)( 2,19,16,10, 7, 4,18,15,12, 6, 3,20,14,11, 8)$ |
$ 15, 5 $ | $4$ | $15$ | $( 1,17,13, 9, 5)( 2,20,15,10, 8, 3,18,16,11, 6, 4,19,14,12, 7)$ |
$ 10, 10 $ | $3$ | $10$ | $( 1,18,13,10, 5, 2,17,14, 9, 6)( 3,20,15,12, 7, 4,19,16,11, 8)$ |
Group invariants
Order: | $60=2^{2} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [60, 9] |
Character table: |
2 2 . . 2 2 . . 2 2 . . 2 2 . . 2 2 . . 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 1 1 1 1 1a 3a 3b 2a 5a 15a 15b 10a 5b 15c 15d 10b 5c 15e 15f 10c 5d 15g 15h 2P 1a 3b 3a 1a 5b 15d 15c 5b 5d 15h 15g 5d 5a 15b 15a 5a 5c 15f 15e 3P 1a 1a 1a 2a 5c 5c 5c 10c 5a 5a 5a 10a 5d 5d 5d 10d 5b 5b 5b 5P 1a 3b 3a 2a 1a 3b 3a 2a 1a 3b 3a 2a 1a 3b 3a 2a 1a 3b 3a 7P 1a 3a 3b 2a 5b 15c 15d 10b 5d 15g 15h 10d 5a 15a 15b 10a 5c 15e 15f 11P 1a 3b 3a 2a 5a 15b 15a 10a 5b 15d 15c 10b 5c 15f 15e 10c 5d 15h 15g 13P 1a 3a 3b 2a 5c 15e 15f 10c 5a 15a 15b 10a 5d 15g 15h 10d 5b 15c 15d X.1 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 B B B B C C C C /C /C /C /C /B /B /B X.3 1 1 1 1 C C C C /B /B /B /B B B B B /C /C /C X.4 1 1 1 1 /C /C /C /C B B B B /B /B /B /B C C C X.5 1 1 1 1 /B /B /B /B /C /C /C /C C C C C B B B X.6 1 A /A 1 1 A /A 1 1 A /A 1 1 A /A 1 1 A /A X.7 1 /A A 1 1 /A A 1 1 /A A 1 1 /A A 1 1 /A A X.8 1 A /A 1 B F /I B C G /H C /C H /G /C /B I /F X.9 1 A /A 1 C G /H C /B I /F /B B F /I B /C H /G X.10 1 A /A 1 /C H /G /C B F /I B /B I /F /B C G /H X.11 1 A /A 1 /B I /F /B /C H /G /C C G /H C B F /I X.12 1 /A A 1 B /I F B C /H G C /C /G H /C /B /F I X.13 1 /A A 1 C /H G C /B /F I /B B /I F B /C /G H X.14 1 /A A 1 /C /G H /C B /I F B /B /F I /B C /H G X.15 1 /A A 1 /B /F I /B /C /G H /C C /H G C B /I F X.16 3 . . -1 3 . . -1 3 . . -1 3 . . -1 3 . . X.17 3 . . -1 D . . -/C /E . . -B E . . -/B /D . . X.18 3 . . -1 E . . -/B D . . -/C /D . . -C /E . . X.19 3 . . -1 /E . . -B /D . . -C D . . -/C E . . X.20 3 . . -1 /D . . -C E . . -/B /E . . -B D . . 2 2 3 . 5 1 10d 2P 5c 3P 10b 5P 2a 7P 10c 11P 10d 13P 10b X.1 1 X.2 /B X.3 /C X.4 C X.5 B X.6 1 X.7 1 X.8 /B X.9 /C X.10 C X.11 B X.12 /B X.13 /C X.14 C X.15 B X.16 -1 X.17 -C X.18 -B X.19 -/B X.20 -/C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(5)^4 C = E(5)^3 D = 3*E(5)^2 E = 3*E(5) F = E(15)^7 G = E(15)^4 H = E(15) I = E(15)^13 |