Properties

Label 25T33
Order \(500\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5^2:F_5$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $33$
Group :  $C_5^2:F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,12,17,25)(2,9,13,18,21)(3,10,14,19,22)(4,6,15,20,23)(5,7,11,16,24), (1,18,21,6)(2,17,22,10)(3,16,23,9)(4,20,24,8)(5,19,25,7)(11,13)(14,15), (1,8,5,7,4,6,3,10,2,9)(11,22,15,21,14,25,13,24,12,23)(16,18,20,17,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$ x 6
100:  25T9

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Low degree siblings

25T33 x 5

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

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [500, 23]
Character table:   
      2  2  .  2  2  2  1   1  1   1  .  .  .  .  .
      5  3  2  1  .  .  3   1  3   1  2  2  2  2  2

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

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

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