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Properties

Label 25T31
Order \(400\)
n \(25\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_5^2.C_4$

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Group action invariants

Degree $n$ :  $25$
Transitive number $t$ :  $31$
Group :  $D_5^2.C_4$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,19,5,7,18,25,6,13,24)(2,17,20,10,8,23,21,11,14,4)(3,22,16,15,9), (1,21,18,8,22,2,10,20)(3,19,17,24,25,9,6,4)(5,12,16,15,23,11,7,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
16:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T28, 20T104, 20T107, 20T109, 20T115, 40T397, 40T398, 40T399, 40T400

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 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $10$ $2$ $( 2, 6)( 3,11)( 4,16)( 5,21)( 8,12)( 9,17)(10,22)(14,18)(15,23)(20,24)$
$ 8, 8, 8, 1 $ $50$ $8$ $( 2, 8, 4,17, 5,24, 3,15)( 6,20,16,23,21,12,11, 9)( 7,22,19,14,25,10,13,18)$
$ 8, 8, 8, 1 $ $50$ $8$ $( 2, 9, 4,20, 5,23, 3,12)( 6,15,16, 8,21,17,11,24)( 7,18,19,22,25,14,13,10)$
$ 4, 4, 4, 4, 4, 4, 1 $ $50$ $4$ $( 2,11, 5,16)( 3,21, 4, 6)( 7,13,25,19)( 8,23,24, 9)(10,18,22,14)(12,15,20,17)$
$ 8, 8, 8, 1 $ $50$ $8$ $( 2,12, 3,23, 5,20, 4, 9)( 6,24,11,17,21, 8,16,15)( 7,10,13,14,25,22,19,18)$
$ 8, 8, 8, 1 $ $50$ $8$ $( 2,15, 3,24, 5,17, 4, 8)( 6, 9,11,12,21,23,16,20)( 7,18,13,10,25,14,19,22)$
$ 5, 5, 5, 5, 5 $ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 10, 10, 5 $ $40$ $10$ $( 1, 2, 7, 8,13,14,19,20,25,21)( 3,12, 9,18,15,24,16, 5,22, 6)( 4,17,10,23,11)$
$ 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:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 206]
Character table:    
      2  4  4  4  4  3  3  3  3  3  3  .   1  1
      5  2  .  .  .  1  .  .  .  .  .  2   1  2

        1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 5a 10a 5b
     2P 1a 2a 2a 1a 1a 4b 4b 2a 4a 4a 5a  5b 5b
     3P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b
     5P 1a 4a 4b 2a 2b 8a 8b 4c 8c 8d 1a  2b 1a
     7P 1a 4b 4a 2a 2b 8d 8c 4c 8b 8a 5a 10a 5b

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

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