Properties

Label 30T32
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times F_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $32$
Group :  $S_3\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,12)(2,21,11)(3,5,20,28,24,26,10,18,14,16,29,7)(4,6,19,27,23,25,9,17,13,15,30,8), (1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
20:  $F_5$
24:  $S_3 \times C_4$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $S_3$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

Low degree siblings

15T11, 30T23, 30T24

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 36]
Character table:   
      2  3  3  3  3  3  3  3  3   .  2   2   2   1  1  2
      3  1  1  1  1  .  .  .  .   1  1   1   1   .  1  1
      5  1  .  .  .  .  .  1  .   1  .   .   .   1  1  1

        1a 2a 4a 4b 4c 4d 2b 2c 15a 6a 12a 12b 10a 5a 3a
     2P 1a 1a 2a 2a 2a 2a 1a 1a 15a 3a  6a  6a  5a 5a 3a
     3P 1a 2a 4b 4a 4d 4c 2b 2c  5a 2a  4b  4a 10a 5a 1a
     5P 1a 2a 4a 4b 4c 4d 2b 2c  3a 6a 12a 12b  2b 1a 3a
     7P 1a 2a 4b 4a 4d 4c 2b 2c 15a 6a 12b 12a 10a 5a 3a
    11P 1a 2a 4b 4a 4d 4c 2b 2c 15a 6a 12b 12a 10a 5a 3a
    13P 1a 2a 4a 4b 4c 4d 2b 2c 15a 6a 12a 12b 10a 5a 3a

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 -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
X.5      1 -1  A -A  A -A  1 -1   1 -1   A  -A   1  1  1
X.6      1 -1 -A  A -A  A  1 -1   1 -1  -A   A   1  1  1
X.7      1 -1  A -A -A  A -1  1   1 -1   A  -A  -1  1  1
X.8      1 -1 -A  A  A -A -1  1   1 -1  -A   A  -1  1  1
X.9      2  2 -2 -2  .  .  .  .  -1 -1   1   1   .  2 -1
X.10     2  2  2  2  .  .  .  .  -1 -1  -1  -1   .  2 -1
X.11     2 -2  B -B  .  .  .  .  -1  1  -A   A   .  2 -1
X.12     2 -2 -B  B  .  .  .  .  -1  1   A  -A   .  2 -1
X.13     4  .  .  .  .  . -4  .  -1  .   .   .   1 -1  4
X.14     4  .  .  .  .  .  4  .  -1  .   .   .  -1 -1  4
X.15     8  .  .  .  .  .  .  .   1  .   .   .   . -2 -4

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i