Properties

Label 25T12
Order \(100\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5^2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$ x 2
20:  $D_{10}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$ x 2

Low degree siblings

10T9 x 2, 20T28 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 $ $5$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 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 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 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 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)$
$ 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 $ $2$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 10, 10, 5 $ $10$ $10$ $( 1, 6,11,16,21)( 2,10,12,20,22, 5, 7,15,17,25)( 3, 9,13,19,23, 4, 8,14,18,24)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$
$ 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,21, 6,16)( 2,12,22, 7,17)( 3,13,23, 8,18)( 4,14,24, 9,19) ( 5,15,25,10,20)$
$ 10, 10, 5 $ $10$ $10$ $( 1,11,21, 6,16)( 2,15,22,10,17, 5,12,25, 7,20)( 3,14,23, 9,18, 4,13,24, 8,19)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,12,23, 9,20)( 2,13,24,10,16)( 3,14,25, 6,17)( 4,15,21, 7,18) ( 5,11,22, 8,19)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,13,25, 7,19)( 2,14,21, 8,20)( 3,15,22, 9,16)( 4,11,23,10,17) ( 5,12,24, 6,18)$

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [100, 13]
Character table:   
      2  2  2  2  2  1   1  1   1  1   1  .  .  1   1  .  .
      5  2  1  1  .  2   1  2   1  2   1  2  2  2   1  2  2

        1a 2a 2b 2c 5a 10a 5b 10b 5c 10c 5d 5e 5f 10d 5g 5h
     2P 1a 1a 1a 1a 5b  5b 5a  5a 5f  5f 5h 5g 5c  5c 5e 5d
     3P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d
     5P 1a 2a 2b 2c 1a  2a 1a  2a 1a  2b 1a 1a 1a  2b 1a 1a
     7P 1a 2a 2b 2c 5b 10b 5a 10a 5f 10d 5h 5g 5c 10c 5e 5d

X.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
X.3      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
X.5      2 -2  .  .  A  -A *A -*A  2   .  A *A  2   .  A *A
X.6      2 -2  .  . *A -*A  A  -A  2   . *A  A  2   . *A  A
X.7      2  . -2  .  2   .  2   .  A  -A  A  A *A -*A *A *A
X.8      2  . -2  .  2   .  2   . *A -*A *A *A  A  -A  A  A
X.9      2  .  2  .  2   .  2   .  A   A  A  A *A  *A *A *A
X.10     2  .  2  .  2   .  2   . *A  *A *A *A  A   A  A  A
X.11     2  2  .  .  A   A *A  *A  2   .  A *A  2   .  A *A
X.12     2  2  .  . *A  *A  A   A  2   . *A  A  2   . *A  A
X.13     4  .  .  .  B   . *B   .  B   .  C -1 *B   . -1 *C
X.14     4  .  .  . *B   .  B   . *B   . *C -1  B   . -1  C
X.15     4  .  .  .  B   . *B   . *B   . -1 *C  B   .  C -1
X.16     4  .  .  . *B   .  B   .  B   . -1  C *B   . *C -1

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
C = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
  = (3+Sqrt(5))/2 = 2+b5