Properties

Label 30T14
Order \(60\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{30}$

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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

30T14

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 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