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Properties

Label 25T18
Order \(200\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times F_5$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $18$
Group :  $D_5\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,5)(6,17,21,12)(7,16,22,11)(8,20,23,15)(9,19,24,14)(10,18,25,13), (1,3)(4,5)(6,23)(7,22)(8,21)(9,25)(10,24)(11,18)(12,17)(13,16)(14,20)(15,19), (1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
10:  $D_{5}$
20:  $F_5$, $D_{10}$
40:  $F_{5}\times C_2$, 20T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $F_5$

Low degree siblings

20T51, 40T168

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

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [200, 41]
Character table:    
      2  3  3  3  3  3  3  3  3  2   2   2   2  2   2   2   2  1   1   .   .
      5  2  1  1  1  1  .  .  .  2   1   1   1  2   1   1   1  2   1   2   2

        1a 4a 4b 2a 2b 4c 4d 2c 5a 20a 20b 10a 5b 20c 20d 10b 5c 10c  5d  5e
     2P 1a 2a 2a 1a 1a 2a 2a 1a 5b 10b 10b  5b 5a 10a 10a  5a 5c  5c  5e  5d
     3P 1a 4b 4a 2a 2b 4d 4c 2c 5b 20d 20c 10b 5a 20b 20a 10a 5c 10c  5e  5d
     5P 1a 4a 4b 2a 2b 4c 4d 2c 1a  4a  4b  2a 1a  4a  4b  2a 1a  2b  1a  1a
     7P 1a 4b 4a 2a 2b 4d 4c 2c 5b 20d 20c 10b 5a 20b 20a 10a 5c 10c  5e  5d
    11P 1a 4b 4a 2a 2b 4d 4c 2c 5a 20b 20a 10a 5b 20d 20c 10b 5c 10c  5d  5e
    13P 1a 4a 4b 2a 2b 4c 4d 2c 5b 20c 20d 10b 5a 20a 20b 10a 5c 10c  5e  5d
    17P 1a 4a 4b 2a 2b 4c 4d 2c 5b 20c 20d 10b 5a 20a 20b 10a 5c 10c  5e  5d
    19P 1a 4b 4a 2a 2b 4d 4c 2c 5a 20b 20a 10a 5b 20d 20c 10b 5c 10c  5d  5e

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
D = 4*E(5)^2+4*E(5)^3
  = -2-2*Sqrt(5) = -2-2r5
E = -E(20)^13-E(20)^17
F = -E(20)-E(20)^9