Properties

Label 25T16
Order \(150\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5^2:S_3$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $16$
Group :  $C_5^2:S_3$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,8)(2,6,13)(3,5,18)(4,24,23)(9,25,17)(10,19,22)(11,14,21)(15,20,16), (1,2,3,4,5)(6,21,8,23,10,25,7,22,9,24)(11,20,13,17,15,19,12,16,14,18)
$|\Aut(F/K)|$:  $1$

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 5: None

Low degree siblings

15T13, 15T14, 30T37, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $50$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $15$ $10$ $( 1, 2, 3, 4, 5)( 6,21, 8,23,10,25, 7,22, 9,24)(11,20,13,17,15,19,12,16,14,18)$
$ 10, 10, 5 $ $15$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $15$ $10$ $( 1, 3,13,15,25,22, 7, 9,19,16)( 2, 8,14,20,21)( 4,18,11, 5,23,12,10,24,17, 6)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 10, 10, 5 $ $15$ $10$ $( 1, 4, 2, 5, 3)( 6,23, 7,24, 8,25, 9,21,10,22)(11,17,12,18,13,19,14,20,15,16)$
$ 5, 5, 5, 5, 5 $ $3$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 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, 5]
Character table:   
      2  1  1  .  1   1   1  1   1  1   1  1  .  .
      3  1  .  1  .   .   .  .   .  .   .  .  .  .
      5  2  1  .  2   1   1  2   1  2   1  2  2  2

        1a 2a 3a 5a 10a 10b 5b 10c 5c 10d 5d 5e 5f
     2P 1a 1a 3a 5b  5b  5d 5d  5c 5a  5a 5c 5f 5e
     3P 1a 2a 1a 5c 10d 10a 5a 10b 5d 10c 5b 5f 5e
     5P 1a 2a 3a 1a  2a  2a 1a  2a 1a  2a 1a 1a 1a
     7P 1a 2a 3a 5b 10b 10c 5d 10d 5a 10a 5c 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  . -1  2   .   .  2   .  2   .  2  2  2
X.4      3 -1  .  A   D  /E /B  /D  B   E /A  F *F
X.5      3 -1  .  B   E   D  A  /E /A  /D /B *F  F
X.6      3 -1  . /A  /D   E  B   D /B  /E  A  F *F
X.7      3 -1  . /B  /E  /D /A   E  A   D  B *F  F
X.8      3  1  .  A  -D -/E /B -/D  B  -E /A  F *F
X.9      3  1  .  B  -E  -D  A -/E /A -/D /B *F  F
X.10     3  1  . /A -/D  -E  B  -D /B -/E  A  F *F
X.11     3  1  . /B -/E -/D /A  -E  A  -D  B *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)^4
E = -E(5)^2
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