Properties

Label 25T5
Order \(50\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:D_5$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$ x 6

Low degree siblings

There are no siblings 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 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,10,14,18,22)( 2, 6,15,19,23)( 3, 7,11,20,24)( 4, 8,12,16,25) ( 5, 9,13,17,21)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,21, 6,16)( 2,12,22, 7,17)( 3,13,23, 8,18)( 4,14,24, 9,19) ( 5,15,25,10,20)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,13,25, 7,19)( 2,14,21, 8,20)( 3,15,22, 9,16)( 4,11,23,10,17) ( 5,12,24, 6,18)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,14,22,10,18)( 2,15,23, 6,19)( 3,11,24, 7,20)( 4,12,25, 8,16) ( 5,13,21, 9,17)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,15,24, 8,17)( 2,11,25, 9,18)( 3,12,21,10,19)( 4,13,22, 6,20) ( 5,14,23, 7,16)$

Group invariants

Order:  $50=2 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [50, 4]
Character table:   
      2  1  1  .  .  .  .  .  .  .  .  .  .  .  .
      5  2  .  2  2  2  2  2  2  2  2  2  2  2  2

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

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      2  .  2  2  A  A  A  A  A *A *A *A *A *A
X.4      2  .  2  2 *A *A *A *A *A  A  A  A  A  A
X.5      2  .  A *A  2  A *A *A  A  2  A *A *A  A
X.6      2  . *A  A  2 *A  A  A *A  2 *A  A  A *A
X.7      2  .  A *A  A  2  A *A *A *A  A  2  A *A
X.8      2  . *A  A *A  2 *A  A  A  A *A  2 *A  A
X.9      2  .  A *A  A *A *A  A  2 *A *A  A  2  A
X.10     2  . *A  A *A  A  A *A  2  A  A *A  2 *A
X.11     2  .  A *A *A  A  2  A *A  A *A *A  A  2
X.12     2  . *A  A  A *A  2 *A  A *A  A  A *A  2
X.13     2  .  A *A *A *A  A  2  A  A  2  A *A *A
X.14     2  . *A  A  A  A *A  2 *A *A  2 *A  A  A

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5