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Properties

Label 25T20
Order \(200\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_5:D_5.C_4$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $20$
Group :  $C_5:D_5.C_4$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,23,4,12,5,9,3,20)(6,17,16,24,21,15,11,8)(7,14,19,10,25,18,13,22), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T18 x 3, 20T56 x 3, 40T171 x 3

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

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [200, 40]
Character table:    
      2  3  3  3  3   3   3   3   3  .  .  .
      5  2  .  .  .   .   .   .   .  2  2  2

        1a 4a 4b 2a  8a  8b  8c  8d 5a 5b 5c
     2P 1a 2a 2a 1a  4b  4a  4a  4b 5a 5b 5c
     3P 1a 4b 4a 2a  8c  8d  8a  8b 5a 5b 5c
     5P 1a 4a 4b 2a  8d  8c  8b  8a 1a 1a 1a
     7P 1a 4b 4a 2a  8b  8a  8d  8c 5a 5b 5c

X.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
X.3      1 -1 -1  1   A  -A  -A   A  1  1  1
X.4      1 -1 -1  1  -A   A   A  -A  1  1  1
X.5      1  A -A -1   B  /B -/B  -B  1  1  1
X.6      1  A -A -1  -B -/B  /B   B  1  1  1
X.7      1 -A  A -1 -/B  -B   B  /B  1  1  1
X.8      1 -A  A -1  /B   B  -B -/B  1  1  1
X.9      8  .  .  .   .   .   .   .  3 -2 -2
X.10     8  .  .  .   .   .   .   . -2 -2  3
X.11     8  .  .  .   .   .   .   . -2  3 -2

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)