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Properties

Label 25T3
Order \(50\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times D_5$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $3$
Group :  $C_5\times D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,5,6,4,10,3,9,2,8)(11,24,15,23,14,22,13,21,12,25)(16,18,20,17,19), (1,20)(2,16)(3,17)(4,18)(5,19)(6,11)(7,12)(8,13)(9,14)(10,15)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $D_{5}$, $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $D_{5}$

Low degree siblings

10T6 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, 1, 1, 1, 1, 1 $ $5$ $2$ $( 6,22)( 7,23)( 8,24)( 9,25)(10,21)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 2, 3, 4, 5)( 6,23, 8,25,10,22, 7,24, 9,21)(11,17,13,19,15,16,12,18,14,20)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 3, 5, 2, 4)( 6,24,10,23, 9,22, 8,21, 7,25)(11,18,15,17,14,16,13,20,12,19)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 4, 2, 5, 3)( 6,25, 7,21, 8,22, 9,23,10,24)(11,19,12,20,13,16,14,17,15,18)$
$ 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)$
$ 10, 10, 5 $ $5$ $10$ $( 1, 5, 4, 3, 2)( 6,21, 9,24, 7,22,10,25, 8,23)(11,20,14,18,12,16,15,19,13,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,12,18,25)( 2, 7,13,19,21)( 3, 8,14,20,22)( 4, 9,15,16,23) ( 5,10,11,17,24)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,14,16,24)( 2, 8,15,17,25)( 3, 9,11,18,21)( 4,10,12,19,22) ( 5, 6,13,20,23)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 8,11,19,23)( 2, 9,12,20,24)( 3,10,13,16,25)( 4, 6,14,17,21) ( 5, 7,15,18,22)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 9,13,17,22)( 2,10,14,18,23)( 3, 6,15,19,24)( 4, 7,11,20,25) ( 5, 8,12,16,21)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,10,15,20,21)( 2, 6,11,16,22)( 3, 7,12,17,23)( 4, 8,13,18,24) ( 5, 9,14,19,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,23, 8,19)( 2,12,24, 9,20)( 3,13,25,10,16)( 4,14,21, 6,17) ( 5,15,22, 7,18)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,25, 6,18)( 2,13,21, 7,19)( 3,14,22, 8,20)( 4,15,23, 9,16) ( 5,11,24,10,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,13,22, 9,17)( 2,14,23,10,18)( 3,15,24, 6,19)( 4,11,25, 7,20) ( 5,12,21, 8,16)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,14,24, 7,16)( 2,15,25, 8,17)( 3,11,21, 9,18)( 4,12,22,10,19) ( 5,13,23, 6,20)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,15,21,10,20)( 2,11,22, 6,16)( 3,12,23, 7,17)( 4,13,24, 8,18) ( 5,14,25, 9,19)$

Group invariants

Order:  $50=2 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [50, 3]
Character table:    
      2  1  1  1   1  1   1  1   1  1   1  .  .  .  .  .  .  .  .  .  .
      5  2  1  2   1  2   1  2   1  2   1  2  2  2  2  2  2  2  2  2  2

        1a 2a 5a 10a 5b 10b 5c 10c 5d 10d 5e 5f 5g 5h 5i 5j 5k 5l 5m 5n
     2P 1a 1a 5b  5b 5d  5d 5a  5a 5c  5c 5k 5m 5j 5l 5n 5f 5h 5e 5g 5i
     3P 1a 2a 5c 10c 5a 10a 5d 10d 5b 10b 5l 5j 5m 5k 5n 5g 5e 5h 5f 5i
     5P 1a 2a 1a  2a 1a  2a 1a  2a 1a  2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a
     7P 1a 2a 5b 10b 5d 10d 5a 10a 5c 10c 5k 5m 5j 5l 5n 5f 5h 5e 5g 5i

X.1      1  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  1  1
X.3      1 -1  A  -A  B  -B /B -/B /A -/A  A  B /B /A  1  A  B /B /A  1
X.4      1 -1  B  -B /A -/A  A  -A /B -/B  B /A  A /B  1  B /A  A /B  1
X.5      1 -1 /B -/B  A  -A /A -/A  B  -B /B  A /A  B  1 /B  A /A  B  1
X.6      1 -1 /A -/A /B -/B  B  -B  A  -A /A /B  B  A  1 /A /B  B  A  1
X.7      1  1  A   A  B   B /B  /B /A  /A  A  B /B /A  1  A  B /B /A  1
X.8      1  1  B   B /A  /A  A   A /B  /B  B /A  A /B  1  B /A  A /B  1
X.9      1  1 /B  /B  A   A /A  /A  B   B /B  A /A  B  1 /B  A /A  B  1
X.10     1  1 /A  /A /B  /B  B   B  A   A /A /B  B  A  1 /A /B  B  A  1
X.11     2  .  2   .  2   .  2   .  2   .  E  E  E  E  E *E *E *E *E *E
X.12     2  .  2   .  2   .  2   .  2   . *E *E *E *E *E  E  E  E  E  E
X.13     2  .  C   .  D   . /D   . /C   .  F  H /H /F  E  I  G /G /I *E
X.14     2  . /C   . /D   .  D   .  C   . /F /H  H  F  E /I /G  G  I *E
X.15     2  .  D   . /C   .  C   . /D   .  G /I  I /G *E  H /F  F /H  E
X.16     2  . /D   .  C   . /C   .  D   . /G  I /I  G *E /H  F /F  H  E
X.17     2  .  D   . /C   .  C   . /D   .  H /F  F /H  E  G /I  I /G *E
X.18     2  . /D   .  C   . /C   .  D   . /H  F /F  H  E /G  I /I  G *E
X.19     2  .  C   .  D   . /D   . /C   .  I  G /G /I *E  F  H /H /F  E
X.20     2  . /C   . /D   .  D   .  C   . /I /G  G  I *E /F /H  H  F  E

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