Properties

Label 30T11
Order \(60\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times A_4$

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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)
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)
$|\Aut(F/K)|$:  $10$

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

20T14

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, 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)