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Properties

Label 30T35
Order \(150\)
n \(30\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_6$

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Group action invariants

Degree $n$ :  $30$
Transitive number $t$ :  $35$
Group :  $C_5^2:C_6$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,10,15,21,27)(4,9,16,22,28)(5,29,23,17,11)(6,30,24,18,12), (1,18,22,8,11,27)(2,17,21,7,12,28)(3,25,24,16,14,5)(4,26,23,15,13,6)(9,20,29,10,19,30)
$|\Aut(F/K)|$:  $10$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: None

Degree 6: $C_6$

Degree 10: None

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

Low degree siblings

15T12 x 2, 25T15, 30T35

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

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [150, 6]
Character table:    
      2  1  .  .  1   1   1   1   1  .  .
      3  1  .  .  1   1   1   1   1  .  .
      5  2  2  2  .   .   .   .   .  2  2

        1a 5a 5b 2a  6a  3a  3b  6b 5c 5d
     2P 1a 5b 5a 1a  3b  3b  3a  3a 5d 5c
     3P 1a 5b 5a 2a  2a  1a  1a  2a 5d 5c
     5P 1a 1a 1a 2a  6b  3b  3a  6a 1a 1a

X.1      1  1  1  1   1   1   1   1  1  1
X.2      1  1  1 -1  -1   1   1  -1  1  1
X.3      1  1  1 -1   C  -C -/C  /C  1  1
X.4      1  1  1 -1  /C -/C  -C   C  1  1
X.5      1  1  1  1 -/C -/C  -C  -C  1  1
X.6      1  1  1  1  -C  -C -/C -/C  1  1
X.7      6  A *A  .   .   .   .   .  B *B
X.8      6 *A  A  .   .   .   .   . *B  B
X.9      6  B *B  .   .   .   .   . *A  A
X.10     6 *B  B  .   .   .   .   .  A *A

A = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
B = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4
  = (-3-Sqrt(5))/2 = -2-b5
C = -E(3)
  = (1-Sqrt(-3))/2 = -b3