Properties

Label 10T17
Order \(200\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_5^2 : C_4) : C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $17$
Group :  $(C_5^2 : C_4) : C_2$
CHM label :  $[5^{2}:4]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,9,3)(2,4,8,6), (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10)
$|\Aut(F/K)|$:  $1$

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:  $C_4\times C_2$
20:  $F_5$ x 2
40:  $F_{5}\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T17, 20T54 x 2, 25T19, 40T169 x 2

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 $ $1$ $1$ $()$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)$
$ 4, 4, 2 $ $25$ $4$ $( 1, 2)( 3, 6, 9, 8)( 4, 5,10, 7)$
$ 4, 4, 2 $ $25$ $4$ $( 1, 2)( 3, 8, 9, 6)( 4, 7,10, 5)$
$ 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
$ 10 $ $20$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 10 $ $20$ $10$ $( 1, 2, 3,10, 5, 8, 7, 6, 9, 4)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2,10, 8, 6, 4)$

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [200, 42]
Character table:   
      2  3  3  3  3  .  3  3  3  3   1   1  1  .  1
      5  2  .  .  .  2  1  .  .  1   1   1  2  2  2

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

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

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