Properties

Label 20T26
Order \(100\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5.D_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $26$
Group :  $D_5.D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,20,10)(2,12,19,8)(3,14,18,6)(4,11,17,9)(5,13,16,7), (1,19,5,20,4,16,3,17,2,18)(6,11,9,13,7,15,10,12,8,14)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  $F_5$, 20T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

Low degree siblings

25T11

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

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [100, 10]
Character table:   
      2  2  .  .  .  .  .  1  1  2  2   1   1  2
      5  2  2  2  2  2  2  2  2  .  .   1   1  1

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

X.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
X.3      1  1  1  1  1  1  1  1  E -E  -1  -1 -1
X.4      1  1  1  1  1  1  1  1 -E  E  -1  -1 -1
X.5      2  A *A *A  A  2  A *A  .  .  -A -*A -2
X.6      2 *A  A  A *A  2 *A  A  .  . -*A  -A -2
X.7      2  A *A *A  A  2  A *A  .  .   A  *A  2
X.8      2 *A  A  A *A  2 *A  A  .  .  *A   A  2
X.9      4 -1 -1 -1 -1 -1  4  4  .  .   .   .  .
X.10     4  B /C  C /B -1  D *D  .  .   .   .  .
X.11     4  C  B /B /C -1 *D  D  .  .   .   .  .
X.12     4 /B  C /C  B -1  D *D  .  .   .   .  .
X.13     4 /C /B  B  C -1 *D  D  .  .   .   .  .

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