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Properties

Label 25T15
Order \(150\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:C_6$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $15$
Group :  $C_5^2:C_6$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,16)(2,24,15)(3,4,9)(5,14,22)(6,13,17)(7,18,11)(8,23,10)(20,21,25), (1,11,14,2,17,19)(3,23,24,5,10,9)(6,15,8,22,18,25)(7,16,13,21,12,20)
$|\Aut(F/K)|$:  $1$

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 5: None

Low degree siblings

15T12 x 2, 30T35 x 2

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

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 2a 3a  6a  6b 3b 5a 5b 5c 5d
     2P 1a 1a 3b  3a  3b 3a 5b 5a 5d 5c
     3P 1a 2a 1a  2a  2a 1a 5b 5a 5d 5c
     5P 1a 2a 3b  6b  6a 3a 1a 1a 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  A -/A  -A /A  1  1  1  1
X.4      1 -1 /A  -A -/A  A  1  1  1  1
X.5      1  1  A  /A   A /A  1  1  1  1
X.6      1  1 /A   A  /A  A  1  1  1  1
X.7      6  .  .   .   .  .  B *B  C *C
X.8      6  .  .   .   .  . *B  B *C  C
X.9      6  .  .   .   .  .  C *C *B  B
X.10     6  .  .   .   .  . *C  C  B *B

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