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Properties

Label 25T36
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:C_{10}.C_2$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $36$
Group :  $C_5^2:C_{10}.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,4,6)(2,7,3,9)(5,8)(11,15,12,13)(16,25,19,22)(17,24,18,23)(20,21), (1,23,2,21)(3,24,5,25)(4,22)(6,13)(7,14,10,12)(8,15,9,11)(16,18,19,17)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  $F_5$, 20T2
100:  20T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T39

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $20$ $5$ $( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $125$ $4$ $( 2, 3, 5, 4)( 6,21)( 7,24,10,23)( 8,22, 9,25)(11,16)(12,19,15,18) (13,17,14,20)$
$ 4, 4, 4, 4, 4, 2, 2, 1 $ $125$ $4$ $( 2, 4, 5, 3)( 6,21)( 7,23,10,24)( 8,25, 9,22)(11,16)(12,18,15,19) (13,20,14,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6,10, 9, 8, 7)(11,14,12,15,13)(16,18,20,17,19) (21,22,23,24,25)$
$ 10, 10, 5 $ $50$ $10$ $( 1, 6,16,11,21)( 2, 7,18,13,22, 5,10,19,14,25)( 3, 8,20,15,23, 4, 9,17,12,24)$
$ 5, 5, 5, 5, 5 $ $10$ $5$ $( 1, 6,16,11,21)( 2,10,18,14,22)( 3, 9,20,12,23)( 4, 8,17,15,24) ( 5, 7,19,13,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,17,13,24)( 2,10,19,11,25)( 3, 9,16,14,21)( 4, 8,18,12,22) ( 5, 7,20,15,23)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,18,15,22)( 2,10,20,13,23)( 3, 9,17,11,24)( 4, 8,19,14,25) ( 5, 7,16,12,21)$
$ 10, 10, 5 $ $50$ $10$ $( 1,11, 6,21,16)( 2,13,10,25,18, 5,14, 7,22,19)( 3,15, 9,24,20, 4,12, 8,23,17)$
$ 5, 5, 5, 5, 5 $ $10$ $5$ $( 1,11, 6,21,16)( 2,14,10,22,18)( 3,12, 9,23,20)( 4,15, 8,24,17) ( 5,13, 7,25,19)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1,11, 8,25,17)( 2,14, 7,21,19)( 3,12, 6,22,16)( 4,15,10,23,18) ( 5,13, 9,24,20)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1,11,10,24,18)( 2,14, 9,25,20)( 3,12, 8,21,17)( 4,15, 7,22,19) ( 5,13, 6,23,16)$

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 21]
Character table:    
      2  2  .  2  2  2  .   1   1   .   .   1   1   .   .
      5  3  2  .  .  1  3   1   2   2   2   1   2   2   2

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

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  A -A -1  1  -1   1   1   1  -1   1   1   1
X.4      1  1 -A  A -1  1  -1   1   1   1  -1   1   1   1
X.5      2  2  .  . -2  2   B  -B  -B  -B  *B -*B -*B -*B
X.6      2  2  .  . -2  2  *B -*B -*B -*B   B  -B  -B  -B
X.7      2  2  .  .  2  2 -*B -*B -*B -*B  -B  -B  -B  -B
X.8      2  2  .  .  2  2  -B  -B  -B  -B -*B -*B -*B -*B
X.9      4 -1  .  .  .  4   .   4  -1  -1   .   4  -1  -1
X.10     4 -1  .  .  .  4   .   C   D  /D   .  *C  /E   E
X.11     4 -1  .  .  .  4   .   C  /D   D   .  *C   E  /E
X.12     4 -1  .  .  .  4   .  *C   E  /E   .   C   D  /D
X.13     4 -1  .  .  .  4   .  *C  /E   E   .   C  /D   D
X.14    20  .  .  .  . -5   .   .   .   .   .   .   .   .

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