Properties

Label 10T33
Order \(800\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_5 \wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $33$
Group :  $F_5 \wr C_2$
CHM label :  $[F(5)^{2}]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,4,8,6)
$|\Aut(F/K)|$:  $1$

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:  $D_{4}$ x 2, $C_4\times C_2$
16:  $C_2^2:C_4$
32:  $C_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

20T155, 20T161, 20T167, 20T169, 25T50, 40T874, 40T875, 40T876, 40T877, 40T878, 40T879, 40T880, 40T881, 40T882, 40T883

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 $ $1$ $1$ $()$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 5, 5 $ $16$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 10 $ $80$ $10$ $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$
$ 4, 4, 1, 1 $ $50$ $4$ $( 3, 5, 9, 7)( 4, 8,10, 6)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 6, 8)$
$ 5, 2, 2, 1 $ $40$ $10$ $( 1, 3, 5, 7, 9)( 4,10)( 6, 8)$
$ 4, 4, 2 $ $100$ $4$ $( 1, 6, 3, 8)( 2, 7)( 4, 5,10, 9)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$
$ 4, 1, 1, 1, 1, 1, 1 $ $10$ $4$ $( 4, 6,10, 8)$
$ 5, 4, 1 $ $40$ $20$ $( 1, 3, 5, 7, 9)( 4, 6,10, 8)$
$ 4, 2, 2, 1, 1 $ $50$ $4$ $( 3, 9)( 4, 8,10, 6)( 5, 7)$
$ 8, 2 $ $100$ $8$ $( 1, 6, 5,10, 3, 8, 9, 4)( 2, 7)$
$ 4, 1, 1, 1, 1, 1, 1 $ $10$ $4$ $( 4, 8,10, 6)$
$ 5, 4, 1 $ $40$ $20$ $( 1, 3, 5, 7, 9)( 4, 8,10, 6)$
$ 4, 2, 2, 1, 1 $ $50$ $4$ $( 3, 9)( 4, 6,10, 8)( 5, 7)$
$ 8, 2 $ $100$ $8$ $( 1, 6, 9, 4, 3, 8, 5,10)( 2, 7)$

Group invariants

Order:  $800=2^{5} \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [800, 1191]
Character table:   
      2  5  2  1  5  3   1  4  4   2  3  5  5   4   2   4  3   4   2   4  3
      5  2  2  2  .  1   1  .  1   1  .  .  .   1   1   .  .   1   1   .  .

        1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d  4e 20a  4f 8a  4g 20b  4h 8b
     2P 1a 5a 5b 1a 1a  5b 2a 1a  5a 2a 2a 2a  2c 10b  2c 4c  2c 10b  2c 4d
     3P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c  4g 20b  4h 8b  4e 20a  4f 8a
     5P 1a 1a 1a 2a 2b  2b 4a 2c  2c 4b 4c 4d  4e  4e  4f 8a  4g  4g  4h 8b
     7P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c  4g 20b  4h 8b  4e 20a  4f 8a
    11P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c  4g 20b  4h 8b  4e 20a  4f 8a
    13P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d  4e 20a  4f 8a  4g 20b  4h 8b
    17P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4c 4d  4e 20a  4f 8a  4g 20b  4h 8b
    19P 1a 5a 5b 2a 2b 10a 4a 2c 10b 4b 4d 4c  4g 20b  4h 8b  4e 20a  4f 8a

X.1      1  1  1  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  -1  -1  -1  1
X.3      1  1  1  1 -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  -1  -1  -1 -1
X.5      1  1  1  1 -1  -1  1 -1  -1  1 -1 -1   B   B   B -B  -B  -B  -B  B
X.6      1  1  1  1 -1  -1  1 -1  -1  1 -1 -1  -B  -B  -B  B   B   B   B -B
X.7      1  1  1  1  1   1  1 -1  -1 -1 -1 -1   B   B   B  B  -B  -B  -B -B
X.8      1  1  1  1  1   1  1 -1  -1 -1 -1 -1  -B  -B  -B -B   B   B   B  B
X.9      2  2  2  2  .   . -2 -2  -2  .  2  2   .   .   .  .   .   .   .  .
X.10     2  2  2  2  .   . -2  2   2  . -2 -2   .   .   .  .   .   .   .  .
X.11     2  2  2 -2  .   .  .  .   .  .  A -A   C   C  -C  .  /C  /C -/C  .
X.12     2  2  2 -2  .   .  .  .   .  . -A  A  /C  /C -/C  .   C   C  -C  .
X.13     2  2  2 -2  .   .  .  .   .  . -A  A -/C -/C  /C  .  -C  -C   C  .
X.14     2  2  2 -2  .   .  .  .   .  .  A -A  -C  -C   C  . -/C -/C  /C  .
X.15     8  3 -2  .  .   .  .  4  -1  .  .  .  -4   1   .  .  -4   1   .  .
X.16     8  3 -2  .  .   .  .  4  -1  .  .  .   4  -1   .  .   4  -1   .  .
X.17     8  3 -2  .  .   .  . -4   1  .  .  .   D  -B   .  .  -D   B   .  .
X.18     8  3 -2  .  .   .  . -4   1  .  .  .  -D   B   .  .   D  -B   .  .
X.19    16 -4  1  . -4   1  .  .   .  .  .  .   .   .   .  .   .   .   .  .
X.20    16 -4  1  .  4  -1  .  .   .  .  .  .   .   .   .  .   .   .   .  .

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