Properties

Label 25T4
Order \(50\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{25}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

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

Group invariants

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

        1a 2a 5a 5b 25a 25b 25c 25d 25e 25f 25g 25h 25i 25j
     2P 1a 1a 5b 5a 25f 25h 25j 25g 25i 25b 25e 25c 25a 25d
     3P 1a 2a 5b 5a 25g 25i 25f 25h 25j 25e 25c 25a 25d 25b
     5P 1a 2a 1a 1a  5a  5a  5a  5a  5a  5b  5b  5b  5b  5b
     7P 1a 2a 5b 5a 25j 25g 25i 25f 25h 25d 25b 25e 25c 25a
    11P 1a 2a 5a 5b 25d 25e 25a 25b 25c 25g 25h 25i 25j 25f
    13P 1a 2a 5b 5a 25i 25f 25h 25j 25g 25a 25d 25b 25e 25c
    17P 1a 2a 5b 5a 25h 25j 25g 25i 25f 25c 25a 25d 25b 25e
    19P 1a 2a 5a 5b 25e 25a 25b 25c 25d 25i 25j 25f 25g 25h
    23P 1a 2a 5b 5a 25f 25h 25j 25g 25i 25b 25e 25c 25a 25d

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   B   E   F   C   D   K   G   J   I   H
X.6      2  .  A *A   C   D   B   E   F   G   J   I   H   K
X.7      2  .  A *A   D   B   E   F   C   I   H   K   G   J
X.8      2  .  A *A   E   F   C   D   B   J   I   H   K   G
X.9      2  .  A *A   F   C   D   B   E   H   K   G   J   I
X.10     2  . *A  A   G   I   K   J   H   D   F   B   C   E
X.11     2  . *A  A   H   G   I   K   J   C   E   D   F   B
X.12     2  . *A  A   I   K   J   H   G   B   C   E   D   F
X.13     2  . *A  A   J   H   G   I   K   F   B   C   E   D
X.14     2  . *A  A   K   J   H   G   I   E   D   F   B   C

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = E(25)^3+E(25)^22
C = E(25)^8+E(25)^17
D = E(25)^7+E(25)^18
E = E(25)^12+E(25)^13
F = -E(25)^3-E(25)^7-E(25)^8-E(25)^12-E(25)^13-E(25)^17-E(25)^18-E(25)^22
G = E(25)^9+E(25)^16
H = E(25)^4+E(25)^21
I = E(25)^11+E(25)^14
J = -E(25)^4-E(25)^6-E(25)^9-E(25)^11-E(25)^14-E(25)^16-E(25)^19-E(25)^21
K = E(25)^6+E(25)^19