Properties

Label 10T28
Order \(400\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_5^2 : C_8):C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $28$
Group :  $(C_5^2 : C_8):C_2$
CHM label :  $1/2[F(5)^{2}]2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10), (2,8)(4,6), (1,6,7,2,9,4,3,8)(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:  $C_4\times C_2$
16:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

20T104, 20T107, 20T109, 20T115, 25T31, 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 6, 8)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$
$ 4, 4, 1, 1 $ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$
$ 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 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 5, 2, 2, 1 $ $40$ $10$ $( 2, 4, 6, 8,10)( 3, 9)( 5, 7)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 4, 5, 6, 9,10, 7, 8)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 4, 7, 8, 9,10, 5, 6)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 6, 5,10, 9, 8, 7, 4)$
$ 8, 2 $ $50$ $8$ $( 1, 2)( 3, 6, 7, 4, 9, 8, 5,10)$
$ 5, 5 $ $16$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [400, 206]
Character table:   
      2  4  3  4  3  4  4  1   1  3  3  3  3  .
      5  2  1  .  .  .  .  2   1  .  .  .  .  2

        1a 2a 4a 4b 4c 2b 5a 10a 8a 8b 8c 8d 5b
     2P 1a 1a 2b 2b 2b 1a 5a  5a 4a 4c 4a 4c 5b
     3P 1a 2a 4c 4b 4a 2b 5a 10a 8d 8c 8b 8a 5b
     5P 1a 2a 4a 4b 4c 2b 1a  2a 8a 8b 8c 8d 1a
     7P 1a 2a 4c 4b 4a 2b 5a 10a 8d 8c 8b 8a 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  1  1  -1  B -B  B -B  1
X.6      1 -1 -1  1 -1  1  1  -1 -B  B -B  B  1
X.7      1  1 -1 -1 -1  1  1   1  B  B -B -B  1
X.8      1  1 -1 -1 -1  1  1   1 -B -B  B  B  1
X.9      2  .  A  . -A -2  2   .  .  .  .  .  2
X.10     2  . -A  .  A -2  2   .  .  .  .  .  2
X.11     8 -4  .  .  .  .  3   1  .  .  .  . -2
X.12     8  4  .  .  .  .  3  -1  .  .  .  . -2
X.13    16  .  .  .  .  . -4   .  .  .  .  .  1

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