Properties

Label 30T37
Order \(150\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:S_3$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $37$
Group :  $C_5^2:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,25,10,19,15,13,21,7,28)(2,3,26,9,20,16,14,22,8,27)(5,17)(6,18)(11,12)(23,29)(24,30), (1,11,22)(2,12,21)(3,13,23)(4,14,24)(5,15,26)(6,16,25)(7,17,27)(8,18,28)(9,19,30)(10,20,29)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: None

Degree 15: $(C_5^2 : C_3):C_2$

Low degree siblings

15T13, 15T14, 25T16, 30T38

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $5$ $( 3, 9,16,22,27)( 4,10,15,21,28)( 5,29,24,18,12)( 6,30,23,17,11)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $5$ $( 3,16,27, 9,22)( 4,15,28,10,21)( 5,24,12,29,18)( 6,23,11,30,17)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $5$ $( 3,22, 9,27,16)( 4,21,10,28,15)( 5,18,29,12,24)( 6,17,30,11,23)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $5$ $( 3,27,22,16, 9)( 4,28,21,15,10)( 5,12,18,24,29)( 6,11,17,23,30)$
$ 10, 10, 2, 2, 2, 2, 2 $ $15$ $10$ $( 1, 2)( 3, 5, 9,29,16,24,22,18,27,12)( 4, 6,10,30,15,23,21,17,28,11)( 7,26) ( 8,25)(13,20)(14,19)$
$ 10, 10, 2, 2, 2, 2, 2 $ $15$ $10$ $( 1, 2)( 3,12,27,18,22,24,16,29, 9, 5)( 4,11,28,17,21,23,15,30,10, 6)( 7,26) ( 8,25)(13,20)(14,19)$
$ 10, 10, 2, 2, 2, 2, 2 $ $15$ $10$ $( 1, 2)( 3,18,16, 5,27,24, 9,12,22,29)( 4,17,15, 6,28,23,10,11,21,30)( 7,26) ( 8,25)(13,20)(14,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3,24)( 4,23)( 5,22)( 6,21)( 7,26)( 8,25)( 9,18)(10,17)(11,15)(12,16) (13,20)(14,19)(27,29)(28,30)$
$ 10, 10, 2, 2, 2, 2, 2 $ $15$ $10$ $( 1, 2)( 3,29,22,12, 9,24,27, 5,16,18)( 4,30,21,11,10,23,28, 6,15,17)( 7,26) ( 8,25)(13,20)(14,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $50$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,16,17)(14,15,18)(19,22,23) (20,21,24)(25,27,30)(26,28,29)$
$ 5, 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 3, 9,16,22,27)( 4,10,15,21,28) ( 5,24,12,29,18)( 6,23,11,30,17)$
$ 5, 5, 5, 5, 5, 5 $ $6$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 3,16,27, 9,22)( 4,15,28,10,21) ( 5,18,29,12,24)( 6,17,30,11,23)$

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [150, 5]
Character table:   
      2  1  1  1  1  1   1   1   1  1   1  .  .  .
      3  1  .  .  .  .   .   .   .  .   .  1  .  .
      5  2  2  2  2  2   1   1   1  1   1  .  2  2

        1a 5a 5b 5c 5d 10a 10b 10c 2a 10d 3a 5e 5f
     2P 1a 5b 5d 5a 5c  5a  5d  5b 1a  5c 3a 5f 5e
     3P 1a 5c 5a 5d 5b 10d 10c 10a 2a 10b 1a 5f 5e
     5P 1a 1a 1a 1a 1a  2a  2a  2a 2a  2a 3a 1a 1a
     7P 1a 5b 5d 5a 5c 10c 10d 10b 2a 10a 3a 5f 5e

X.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
X.3      2  2  2  2  2   .   .   .  .   . -1  2  2
X.4      3  A /B  B /A   D  /D  /E -1   E  .  F *F
X.5      3  B  A /A /B   E  /E   D -1  /D  . *F  F
X.6      3 /A  B /B  A  /D   D   E -1  /E  .  F *F
X.7      3 /B /A  A  B  /E   E  /D -1   D  . *F  F
X.8      3  A /B  B /A  -D -/D -/E  1  -E  .  F *F
X.9      3  B  A /A /B  -E -/E  -D  1 -/D  . *F  F
X.10     3 /A  B /B  A -/D  -D  -E  1 -/E  .  F *F
X.11     3 /B /A  A  B -/E  -E -/D  1  -D  . *F  F
X.12     6  C *C *C  C   .   .   .  .   .  .  G *G
X.13     6 *C  C  C *C   .   .   .  .   .  . *G  G

A = 2*E(5)^3+E(5)^4
B = E(5)^2+2*E(5)^4
C = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
D = -E(5)^2
E = -E(5)
F = -E(5)^2-E(5)^3
  = (1+Sqrt(5))/2 = 1+b5
G = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4
  = (-3-Sqrt(5))/2 = -2-b5