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Properties

Label 30T17
Order \(120\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:F_5$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $17$
Group :  $C_2\times C_3:F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6,9,18)(4,5,10,17)(7,14,25,19)(8,13,26,20)(11,21)(12,22)(15,29,27,24)(16,30,28,23), (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)
$|\Aut(F/K)|$:  $2$

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:  $D_{6}$, $C_3 : C_4$ x 2
20:  $F_5$
24:  24T6
40:  $F_{5}\times C_2$
60:  $C_{15} : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $D_{6}$

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

Degree 15: $C_{15} : C_4$

Low degree siblings

30T17

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

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 41]
Character table:    
      2  3  3  3  3  3  3  3  3   1  2   1  2  1   1  2  2   1   1
      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

        1a 4a 2a 4b 2b 4c 2c 4d 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b
     2P 1a 2a 1a 2a 1a 2a 1a 2a 15a 3a 15a 3a 5a  5a 3a 3a 15b 15b
     3P 1a 4b 2a 4a 2b 4d 2c 4c 10a 2c  5a 2a 5a 10a 1a 2b  5a 10a
     5P 1a 4a 2a 4b 2b 4c 2c 4d  6c 6a  3a 6b 1a  2b 3a 6c  3a  6c
     7P 1a 4b 2a 4a 2b 4d 2c 4c 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a
    11P 1a 4b 2a 4a 2b 4d 2c 4c 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a
    13P 1a 4a 2a 4b 2b 4c 2c 4d 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a
    17P 1a 4a 2a 4b 2b 4c 2c 4d 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b
    19P 1a 4b 2a 4a 2b 4d 2c 4c 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b
    23P 1a 4b 2a 4a 2b 4d 2c 4c 30a 6a 15a 6b 5a 10a 3a 6c 15b 30b
    29P 1a 4a 2a 4b 2b 4c 2c 4d 30b 6a 15b 6b 5a 10a 3a 6c 15a 30a

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      1  A -1 -A -1 -A  1  A  -1  1   1 -1  1  -1  1 -1   1  -1
X.6      1 -A -1  A -1  A  1 -A  -1  1   1 -1  1  -1  1 -1   1  -1
X.7      1  A -1 -A  1  A -1 -A   1 -1   1 -1  1   1  1  1   1   1
X.8      1 -A -1  A  1 -A -1  A   1 -1   1 -1  1   1  1  1   1   1
X.9      2  . -2  . -2  .  2  .   1 -1  -1  1  2  -2 -1  1  -1   1
X.10     2  . -2  .  2  . -2  .  -1  1  -1  1  2   2 -1 -1  -1  -1
X.11     2  .  2  . -2  . -2  .   1  1  -1 -1  2  -2 -1  1  -1   1
X.12     2  .  2  .  2  .  2  .  -1 -1  -1 -1  2   2 -1 -1  -1  -1
X.13     4  .  .  .  4  .  .  .  -1  .  -1  . -1  -1  4  4  -1  -1
X.14     4  .  .  . -4  .  .  .   1  .  -1  . -1   1  4 -4  -1   1
X.15     4  .  .  . -4  .  .  .   B  .  -B  . -1   1 -2  2 -/B  /B
X.16     4  .  .  . -4  .  .  .  /B  . -/B  . -1   1 -2  2  -B   B
X.17     4  .  .  .  4  .  .  . -/B  . -/B  . -1  -1 -2 -2  -B  -B
X.18     4  .  .  .  4  .  .  .  -B  .  -B  . -1  -1 -2 -2 -/B -/B

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(15)-E(15)^2-E(15)^4-E(15)^8
  = (-1-Sqrt(-15))/2 = -1-b15