Properties

Label 30T7
Order \(60\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times F_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $7$
Group :  $C_3\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,18,25,22,23,7,16,11,14,28,6)(2,3,17,26,21,24,8,15,12,13,27,5)(9,30,20,10,29,19), (1,7,13,20,26)(2,8,14,19,25)(3,9,15,22,28)(4,10,16,21,27)(5,11,18,24,29)(6,12,17,23,30)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: $F_5$

Degree 6: $C_6$

Degree 10: $F_5$

Degree 15: $F_5\times C_3$

Low degree siblings

15T8

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, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 9)( 4,10)( 5,18)( 6,17)( 7,26)( 8,25)(13,20)(14,19)(15,28)(16,27)(23,30) (24,29)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $5$ $4$ $( 1, 2)( 3,16, 9,27)( 4,15,10,28)( 5,30,18,23)( 6,29,17,24)( 7,14,26,19) ( 8,13,25,20)(11,12)(21,22)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ $5$ $4$ $( 1, 2)( 3,27, 9,16)( 4,28,10,15)( 5,23,18,30)( 6,24,17,29)( 7,19,26,14) ( 8,20,25,13)(11,12)(21,22)$
$ 15, 15 $ $4$ $15$ $( 1, 3, 5, 7, 9,11,13,15,18,20,22,24,26,28,29)( 2, 4, 6, 8,10,12,14,16,17,19, 21,23,25,27,30)$
$ 6, 6, 6, 6, 3, 3 $ $5$ $6$ $( 1, 3,11,13,22,24)( 2, 4,12,14,21,23)( 5,20,15,29,26, 9)( 6,19,16,30,25,10) ( 7,28,18)( 8,27,17)$
$ 12, 12, 6 $ $5$ $12$ $( 1, 4,18,25,22,23, 7,16,11,14,28, 6)( 2, 3,17,26,21,24, 8,15,12,13,27, 5) ( 9,30,20,10,29,19)$
$ 12, 12, 6 $ $5$ $12$ $( 1, 4,29, 8,22,23,20,27,11,14, 9,17)( 2, 3,30, 7,21,24,19,28,12,13,10,18) ( 5,25,15, 6,26,16)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 9,13,18,22,26,29, 3, 7,11,15,20,24,28)( 2, 6,10,14,17,21,25,30, 4, 8, 12,16,19,23,27)$
$ 6, 6, 6, 6, 3, 3 $ $5$ $6$ $( 1, 5,22,26,11,15)( 2, 6,21,25,12,16)( 3,13,24)( 4,14,23)( 7,29,28,20,18, 9) ( 8,30,27,19,17,10)$
$ 12, 12, 6 $ $5$ $12$ $( 1, 6,28,14,11,16, 7,23,22,25,18, 4)( 2, 5,27,13,12,15, 8,24,21,26,17, 3) ( 9,19,29,10,20,30)$
$ 12, 12, 6 $ $5$ $12$ $( 1, 6, 3,19,11,16,13,30,22,25,24,10)( 2, 5, 4,20,12,15,14,29,21,26,23, 9) ( 7,17,28, 8,18,27)$
$ 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 7,13,20,26)( 2, 8,14,19,25)( 3, 9,15,22,28)( 4,10,16,21,27) ( 5,11,18,24,29)( 6,12,17,23,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,11,22)( 2,12,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,28) ( 8,17,27)( 9,20,29)(10,19,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,22,11)( 2,21,12)( 3,24,13)( 4,23,14)( 5,26,15)( 6,25,16)( 7,28,18) ( 8,27,17)( 9,29,20)(10,30,19)$

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [60, 6]
Character table:   
      2  2  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  1
      5  1  .  .  .   1   .   .   .   1   .   .   .  1  1  1

        1a 2a 4a 4b 15a  6a 12a 12b 15b  6b 12c 12d 5a 3a 3b
     2P 1a 1a 2a 2a 15b  3a  6b  6b 15a  3b  6a  6a 5a 3b 3a
     3P 1a 2a 4b 4a  5a  2a  4b  4a  5a  2a  4a  4b 5a 1a 1a
     5P 1a 2a 4a 4b  3a  6b 12d 12c  3b  6a 12b 12a 1a 3b 3a
     7P 1a 2a 4b 4a 15a  6a 12b 12a 15b  6b 12d 12c 5a 3a 3b
    11P 1a 2a 4b 4a 15b  6b 12c 12d 15a  6a 12a 12b 5a 3b 3a
    13P 1a 2a 4a 4b 15a  6a 12a 12b 15b  6b 12c 12d 5a 3a 3b

X.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
X.3      1 -1  A -A   1  -1   A  -A   1  -1  -A   A  1  1  1
X.4      1 -1 -A  A   1  -1  -A   A   1  -1   A  -A  1  1  1
X.5      1 -1  A -A   B  -B   C  -C  /B -/B  /C -/C  1 /B  B
X.6      1 -1  A -A  /B -/B -/C  /C   B  -B  -C   C  1  B /B
X.7      1 -1 -A  A   B  -B  -C   C  /B -/B -/C  /C  1 /B  B
X.8      1 -1 -A  A  /B -/B  /C -/C   B  -B   C  -C  1  B /B
X.9      1  1 -1 -1   B   B  -B  -B  /B  /B -/B -/B  1 /B  B
X.10     1  1 -1 -1  /B  /B -/B -/B   B   B  -B  -B  1  B /B
X.11     1  1  1  1   B   B   B   B  /B  /B  /B  /B  1 /B  B
X.12     1  1  1  1  /B  /B  /B  /B   B   B   B   B  1  B /B
X.13     4  .  .  .  -1   .   .   .  -1   .   .   . -1  4  4
X.14     4  .  .  . -/B   .   .   .  -B   .   .   . -1  D /D
X.15     4  .  .  .  -B   .   .   . -/B   .   .   . -1 /D  D

A = -E(4)
  = -Sqrt(-1) = -i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = -E(12)^11
D = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3