Group action invariants
| Degree $n$ : | $30$ | |
| Transitive number $t$ : | $14$ | |
| Group : | $D_{30}$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,30)(2,29)(3,27)(4,28)(5,26)(6,25)(7,23)(8,24)(9,21)(10,22)(11,20)(12,19)(13,17)(14,18)(15,16), (3,30)(4,29)(5,27)(6,28)(7,25)(8,26)(9,23)(10,24)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18) | |
| $|\Aut(F/K)|$: | $2$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_2^2$ 6: $S_3$ 10: $D_{5}$ 12: $D_{6}$ 20: $D_{10}$ 30: $D_{15}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: $D_{5}$
Degree 6: $D_{6}$
Degree 10: $D_{10}$
Degree 15: $D_{15}$
Low degree siblings
30T14Siblings 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, 2, 2, 1, 1 $ | $15$ | $2$ | $( 3,30)( 4,29)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22)(13,20) (14,19)(15,17)(16,18)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,29)( 4,30)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,19)(14,20)(15,18)(16,17)$ |
| $ 30 $ | $2$ | $30$ | $( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19, 22,24,26,27,30)$ |
| $ 15, 15 $ | $2$ | $15$ | $( 1, 4, 5, 7,10,11,14,15,17,19,21,24,25,27,29)( 2, 3, 6, 8, 9,12,13,16,18,20, 22,23,26,28,30)$ |
| $ 15, 15 $ | $2$ | $15$ | $( 1, 5,10,14,17,21,25,29, 4, 7,11,15,19,24,27)( 2, 6, 9,13,18,22,26,30, 3, 8, 12,16,20,23,28)$ |
| $ 30 $ | $2$ | $30$ | $( 1, 6,10,13,17,22,25,30, 4, 8,11,16,19,23,27, 2, 5, 9,14,18,21,26,29, 3, 7, 12,15,20,24,28)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$ |
| $ 10, 10, 10 $ | $2$ | $10$ | $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$ |
| $ 30 $ | $2$ | $30$ | $( 1, 9,17,26, 4,12,19,28, 5,13,21,30, 7,16,24, 2,10,18,25, 3,11,20,27, 6,14, 22,29, 8,15,23)$ |
| $ 15, 15 $ | $2$ | $15$ | $( 1,10,17,25, 4,11,19,27, 5,14,21,29, 7,15,24)( 2, 9,18,26, 3,12,20,28, 6,13, 22,30, 8,16,23)$ |
| $ 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 $ | $2$ | $6$ | $( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$ |
| $ 10, 10, 10 $ | $2$ | $10$ | $( 1,13,25, 8,19, 2,14,26, 7,20)( 3,15,28,10,22, 4,16,27, 9,21)( 5,18,29,12,24, 6,17,30,11,23)$ |
| $ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,14,25, 7,19)( 2,13,26, 8,20)( 3,16,28, 9,22)( 4,15,27,10,21) ( 5,17,29,11,24)( 6,18,30,12,23)$ |
| $ 15, 15 $ | $2$ | $15$ | $( 1,15,29,14,27,11,25,10,24, 7,21, 5,19, 4,17)( 2,16,30,13,28,12,26, 9,23, 8, 22, 6,20, 3,18)$ |
| $ 30 $ | $2$ | $30$ | $( 1,16,29,13,27,12,25, 9,24, 8,21, 6,19, 3,17, 2,15,30,14,28,11,26,10,23, 7, 22, 5,20, 4,18)$ |
Group invariants
| Order: | $60=2^{2} \cdot 3 \cdot 5$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | [60, 12] |
| Character table: |
2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 . 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
1a 2a 2b 2c 30a 15a 15b 30b 5a 10a 30c 15c 3a 6a 10b 5b 15d 30d
2P 1a 1a 1a 1a 15b 15b 15c 15c 5b 5b 15d 15d 3a 3a 5a 5a 15a 15a
3P 1a 2a 2b 2c 10a 5a 5b 10b 5b 10b 10a 5a 1a 2b 10a 5a 5b 10b
5P 1a 2a 2b 2c 6a 3a 3a 6a 1a 2b 6a 3a 3a 6a 2b 1a 3a 6a
7P 1a 2a 2b 2c 30d 15d 15a 30a 5b 10b 30b 15b 3a 6a 10a 5a 15c 30c
11P 1a 2a 2b 2c 30c 15c 15d 30d 5a 10a 30a 15a 3a 6a 10b 5b 15b 30b
13P 1a 2a 2b 2c 30b 15b 15c 30c 5b 10b 30d 15d 3a 6a 10a 5a 15a 30a
17P 1a 2a 2b 2c 30b 15b 15c 30c 5b 10b 30d 15d 3a 6a 10a 5a 15a 30a
19P 1a 2a 2b 2c 30c 15c 15d 30d 5a 10a 30a 15a 3a 6a 10b 5b 15b 30b
23P 1a 2a 2b 2c 30d 15d 15a 30a 5b 10b 30b 15b 3a 6a 10a 5a 15c 30c
29P 1a 2a 2b 2c 30a 15a 15b 30b 5a 10a 30c 15c 3a 6a 10b 5b 15d 30d
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 2 . 2 . -1 -1 -1 -1 2 2 -1 -1 -1 -1 2 2 -1 -1
X.6 2 . -2 . 1 -1 -1 1 2 -2 1 -1 -1 1 -2 2 -1 1
X.7 2 . -2 . A -A -*A *A -*A *A A -A 2 -2 A -A -*A *A
X.8 2 . -2 . *A -*A -A A -A A *A -*A 2 -2 *A -*A -A A
X.9 2 . 2 . -*A -*A -A -A -A -A -*A -*A 2 2 -*A -*A -A -A
X.10 2 . 2 . -A -A -*A -*A -*A -*A -A -A 2 2 -A -A -*A -*A
X.11 2 . -2 . B -B -C C -A A D -D -1 1 *A -*A -E E
X.12 2 . -2 . C -C -D D -*A *A E -E -1 1 A -A -B B
X.13 2 . -2 . D -D -E E -A A B -B -1 1 *A -*A -C C
X.14 2 . -2 . E -E -B B -*A *A C -C -1 1 A -A -D D
X.15 2 . 2 . -E -E -B -B -*A -*A -C -C -1 -1 -A -A -D -D
X.16 2 . 2 . -D -D -E -E -A -A -B -B -1 -1 -*A -*A -C -C
X.17 2 . 2 . -C -C -D -D -*A -*A -E -E -1 -1 -A -A -B -B
X.18 2 . 2 . -B -B -C -C -A -A -D -D -1 -1 -*A -*A -E -E
A = -E(5)-E(5)^4
= (1-Sqrt(5))/2 = -b5
B = -E(15)-E(15)^14
C = -E(15)^2-E(15)^13
D = -E(15)^4-E(15)^11
E = -E(15)^7-E(15)^8
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