Properties

Label 25T19
Order \(200\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5:F_5$

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Group action invariants

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

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:  $C_4\times C_2$
20:  $F_5$ x 2
40:  $F_{5}\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$ x 2

Low degree siblings

10T17 x 2, 20T54 x 2, 40T169 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 $ $5$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 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 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $20$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 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, 42]
Character table:   
      2  3  3  3  3  3  3  3  3  1   1  1   1  .  .
      5  2  1  .  .  .  .  1  .  2   1  2   1  2  2

        1a 2a 4a 4b 4c 4d 2b 2c 5a 10a 5b 10b 5c 5d
     2P 1a 1a 2c 2c 2c 2c 1a 1a 5a  5a 5b  5b 5c 5d
     3P 1a 2a 4d 4c 4b 4a 2b 2c 5a 10a 5b 10b 5c 5d
     5P 1a 2a 4a 4b 4c 4d 2b 2c 1a  2a 1a  2b 1a 1a
     7P 1a 2a 4d 4c 4b 4a 2b 2c 5a 10a 5b 10b 5c 5d

X.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
X.3      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
X.5      1 -1  A -A  A -A  1 -1  1  -1  1   1  1  1
X.6      1 -1 -A  A -A  A  1 -1  1  -1  1   1  1  1
X.7      1  1  A  A -A -A -1 -1  1   1  1  -1  1  1
X.8      1  1 -A -A  A  A -1 -1  1   1  1  -1  1  1
X.9      4  .  .  .  .  . -4  .  4   . -1   1 -1 -1
X.10     4  .  .  .  .  .  4  .  4   . -1  -1 -1 -1
X.11     4 -4  .  .  .  .  .  . -1   1  4   . -1 -1
X.12     4  4  .  .  .  .  .  . -1  -1  4   . -1 -1
X.13     8  .  .  .  .  .  .  . -2   . -2   . -2  3
X.14     8  .  .  .  .  .  .  . -2   . -2   .  3 -2

A = -E(4)
  = -Sqrt(-1) = -i