Properties

Label 25T30
Order \(400\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_5\wr C_2:C_2$

Learn more about

Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $30$
Group :  $D_5\wr C_2:C_2$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,22,11,18,7,14,3,10,24)(2,15,23,6,19)(4,5,25,21,16,17,12,13,8,9), (1,7,25,19)(2,10,24,16)(3,8,23,18)(4,6,22,20)(5,9,21,17)(11,12,15,14), (1,15)(2,25)(3,10)(4,20)(6,13)(7,23)(9,18)(12,21)(14,16)(17,24)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$
16:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T27 x 3, 20T90 x 3, 20T96 x 3, 20T97 x 3, 40T393 x 3, 40T394 x 3, 40T395 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
$ 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 $ $50$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 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 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2, 6, 5,21)( 3,11, 4,16)( 7,10,25,22)( 8,15,24,17)( 9,20,23,12)(13,14,19,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2,11)( 3,21)( 4, 6)( 5,16)( 7,14)( 8,24)(10,19)(13,22)(15,17)(18,25)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2,12,13,23,24, 9,10,20,16)( 3,22,14, 8,25,19, 6, 5,17,11)( 4, 7,15,18,21)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)$
$ 5, 5, 5, 5, 5 $ $8$ $5$ $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)$

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 207]
Character table:   
      2  4  3  4  3  4  4  3  3  3  3  1   1   1   1  1  1
      5  2  1  .  .  .  .  1  .  .  1  2   1   1   1  2  2

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

X.1      1  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  1  1
X.3      1 -1 -1  1 -1  1  1 -1 -1  1  1  -1   1   1  1  1
X.4      1 -1  1 -1  1  1 -1  1 -1  1  1  -1  -1   1  1  1
X.5      1 -1  1 -1  1  1  1 -1  1 -1  1  -1   1  -1  1  1
X.6      1  1 -1 -1 -1  1 -1 -1  1  1  1   1  -1   1  1  1
X.7      1  1 -1 -1 -1  1  1  1 -1 -1  1   1   1  -1  1  1
X.8      1  1  1  1  1  1 -1 -1 -1 -1  1   1  -1  -1  1  1
X.9      2  .  A  . -A -2  .  .  .  .  2   .   .   .  2  2
X.10     2  . -A  .  A -2  .  .  .  .  2   .   .   .  2  2
X.11     8 -4  .  .  .  .  .  .  .  .  3   1   .   . -2 -2
X.12     8  4  .  .  .  .  .  .  .  .  3  -1   .   . -2 -2
X.13     8  .  .  .  .  . -4  .  .  . -2   .   1   .  3 -2
X.14     8  .  .  .  .  .  .  .  . -4 -2   .   .   1 -2  3
X.15     8  .  .  .  .  .  .  .  .  4 -2   .   .  -1 -2  3
X.16     8  .  .  .  .  .  4  .  .  . -2   .  -1   .  3 -2

A = -2*E(4)
  = -2*Sqrt(-1) = -2i