Group action invariants
| Degree $n$: | $30$ | |
| Transitive number $t$: | $11$ | |
| Group: | $C_5\times A_4$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $10$ | |
| Generators: | (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20), (1,20,29,10,17,28,7,15,25,5,14,23,4,11,21)(2,19,30,9,18,27,8,16,26,6,13,24,3,12,22) |
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 3: $C_3$
Degree 5: $C_5$
Degree 6: $A_4$
Degree 10: None
Degree 15: $C_{15}$
Low degree siblings
20T14Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$ |
| $ 10, 10, 5, 5 $ | $3$ | $10$ | $( 1, 3, 5, 8,10, 2, 4, 6, 7, 9)(11,13,15,18,20,12,14,16,17,19)(21,23,25,28,29) (22,24,26,27,30)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 4, 5, 7,10)( 2, 3, 6, 8, 9)(11,14,15,17,20)(12,13,16,18,19) (21,23,25,28,29)(22,24,26,27,30)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 5,10, 4, 7)( 2, 6, 9, 3, 8)(11,15,20,14,17)(12,16,19,13,18) (21,25,29,23,28)(22,26,30,24,27)$ |
| $ 10, 10, 5, 5 $ | $3$ | $10$ | $( 1, 5,10, 4, 7)( 2, 6, 9, 3, 8)(11,16,20,13,17,12,15,19,14,18) (21,26,29,24,28,22,25,30,23,27)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1, 7, 4,10, 5)( 2, 8, 3, 9, 6)(11,17,14,20,15)(12,18,13,19,16) (21,28,23,29,25)(22,27,24,30,26)$ |
| $ 10, 10, 5, 5 $ | $3$ | $10$ | $( 1, 7, 4,10, 5)( 2, 8, 3, 9, 6)(11,18,14,19,15,12,17,13,20,16) (21,27,23,30,25,22,28,24,29,26)$ |
| $ 10, 10, 5, 5 $ | $3$ | $10$ | $( 1, 9, 7, 6, 4, 2,10, 8, 5, 3)(11,19,17,16,14,12,20,18,15,13)(21,29,28,25,23) (22,30,27,26,24)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $1$ | $5$ | $( 1,10, 7, 5, 4)( 2, 9, 8, 6, 3)(11,20,17,15,14)(12,19,18,16,13) (21,29,28,25,23)(22,30,27,26,24)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,11,23, 5,15,28,10,20,21, 4,14,25, 7,17,29)( 2,12,24, 6,16,27, 9,19,22, 3, 13,26, 8,18,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,13,27)( 2,14,28)( 3,15,29)( 4,16,30)( 5,18,22)( 6,17,21)( 7,19,24) ( 8,20,23)( 9,11,25)(10,12,26)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,15,22, 7,11,27, 4,17,24,10,14,30, 5,20,26)( 2,16,21, 8,12,28, 3,18,23, 9, 13,29, 6,19,25)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,17,25, 4,20,28, 5,11,29, 7,14,21,10,15,23)( 2,18,26, 3,19,27, 6,12,30, 8, 13,22, 9,16,24)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,19,30,10,18,27, 7,16,26, 5,13,24, 4,12,22)( 2,20,29, 9,17,28, 8,15,25, 6, 14,23, 3,11,21)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,21,12, 4,23,13, 5,25,16, 7,28,18,10,29,19)( 2,22,11, 3,24,14, 6,26,15, 8, 27,17, 9,30,20)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,23,15,10,21,14, 7,29,11, 5,28,20, 4,25,17)( 2,24,16, 9,22,13, 8,30,12, 6, 27,19, 3,26,18)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,25,20, 5,29,14,10,23,17, 4,28,11, 7,21,15)( 2,26,19, 6,30,13, 9,24,18, 3, 27,12, 8,22,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1,27,13)( 2,28,14)( 3,29,15)( 4,30,16)( 5,22,18)( 6,21,17)( 7,24,19) ( 8,23,20)( 9,25,11)(10,26,12)$ |
| $ 15, 15 $ | $4$ | $15$ | $( 1,29,17, 7,25,14, 4,21,20,10,28,15, 5,23,11)( 2,30,18, 8,26,13, 3,22,19, 9, 27,16, 6,24,12)$ |
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 2 . . . . . . . . .
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 1 1 1 1 1 1 1 1 1
1a 2a 10a 5a 5b 10b 5c 10c 10d 5d 15a 3a 15b 15c 15d 15e 15f 15g 3b
2P 1a 1a 5b 5b 5d 5d 5a 5a 5c 5c 15f 3b 15e 15g 15h 15a 15b 15d 3a
3P 1a 2a 10c 5c 5a 10a 5d 10d 10b 5b 5b 1a 5c 5a 5d 5a 5d 5b 1a
5P 1a 2a 2a 1a 1a 2a 1a 2a 2a 1a 3b 3b 3b 3b 3b 3a 3a 3a 3a
7P 1a 2a 10b 5b 5d 10d 5a 10a 10c 5c 15d 3a 15c 15a 15b 15g 15h 15f 3b
11P 1a 2a 10a 5a 5b 10b 5c 10c 10d 5d 15g 3b 15h 15e 15f 15c 15d 15a 3a
13P 1a 2a 10c 5c 5a 10a 5d 10d 10b 5b 15c 3a 15d 15b 15a 15h 15g 15e 3b
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 1 1 1 1 1 1 E E E E E /E /E /E /E
X.3 1 1 1 1 1 1 1 1 1 1 /E /E /E /E /E E E E E
X.4 1 1 A A B B /B /B /A /A /A 1 A B /B B /B /A 1
X.5 1 1 B B /A /A A A /B /B /B 1 B /A A /A A /B 1
X.6 1 1 /B /B A A /A /A B B B 1 /B A /A A /A B 1
X.7 1 1 /A /A /B /B B B A A A 1 /A /B B /B B A 1
X.8 1 1 A A B B /B /B /A /A F E /G /H I /I H G /E
X.9 1 1 A A B B /B /B /A /A G /E /F /I H /H I F E
X.10 1 1 B B /A /A A A /B /B H /E /I G /F F /G I E
X.11 1 1 B B /A /A A A /B /B I E /H F /G G /F H /E
X.12 1 1 /B /B A A /A /A B B /I /E H /F G /G F /H E
X.13 1 1 /B /B A A /A /A B B /H E I /G F /F G /I /E
X.14 1 1 /A /A /B /B B B A A /G E F I /H H /I /F /E
X.15 1 1 /A /A /B /B B B A A /F /E G H /I I /H /G E
X.16 3 -1 -1 3 3 -1 3 -1 -1 3 . . . . . . . . .
X.17 3 -1 -/B C /D -A D -/A -B /C . . . . . . . . .
X.18 3 -1 -/A D C -/B /C -B -A /D . . . . . . . . .
X.19 3 -1 -A /D /C -B C -/B -/A D . . . . . . . . .
X.20 3 -1 -B /C D -/A /D -A -/B C . . . . . . . . .
2 .
3 1
5 1
15h
2P 15c
3P 5c
5P 3a
7P 15e
11P 15b
13P 15f
X.1 1
X.2 /E
X.3 E
X.4 A
X.5 B
X.6 /B
X.7 /A
X.8 /F
X.9 /G
X.10 /H
X.11 /I
X.12 I
X.13 H
X.14 G
X.15 F
X.16 .
X.17 .
X.18 .
X.19 .
X.20 .
A = E(5)^4
B = E(5)^3
C = 3*E(5)^2
D = 3*E(5)
E = E(3)^2
= (-1-Sqrt(-3))/2 = -1-b3
F = E(15)^13
G = E(15)^8
H = E(15)^11
I = E(15)
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