Properties

Label 10T23
Order \(320\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times (C_2^4 : D_5)$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $23$
Group :  $C_2\times (C_2^4 : D_5)$
CHM label :  $[2^{5}]D(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (5,10), (1,9)(2,8)(3,7)(4,6), (1,3,5,7,9)(2,4,6,8,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$
20:  $D_{10}$
160:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $D_{5}$

Low degree siblings

10T23 x 5, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 3, 20T85 x 6, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 x 6

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 4, 9)( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 8)( 5,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3, 8)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $20$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)$
$ 4, 2, 2, 1, 1 $ $20$ $4$ $( 2, 5, 7,10)( 3, 4)( 8, 9)$
$ 4, 2, 2, 1, 1 $ $20$ $4$ $( 2, 5)( 3, 4, 8, 9)( 7,10)$
$ 4, 4, 1, 1 $ $20$ $4$ $( 2, 5, 7,10)( 3, 4, 8, 9)$
$ 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 2)( 3, 5)( 4, 9)( 6, 7)( 8,10)$
$ 4, 2, 2, 2 $ $20$ $4$ $( 1, 2)( 3, 5, 8,10)( 4, 9)( 6, 7)$
$ 5, 5 $ $32$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
$ 10 $ $32$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 4, 2, 2, 2 $ $20$ $4$ $( 1, 2, 6, 7)( 3, 5)( 4, 9)( 8,10)$
$ 4, 4, 2 $ $20$ $4$ $( 1, 2, 6, 7)( 3, 5, 8,10)( 4, 9)$
$ 5, 5 $ $32$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$
$ 10 $ $32$ $10$ $( 1, 3, 5, 7, 9, 6, 8,10, 2, 4)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1636]
Character table:   
      2  6  6  6  6  6  4  4  4  4  6  6  4  4  1   1  4  4  1   1  6
      5  1  .  .  .  .  .  .  .  .  .  .  .  .  1   1  .  .  1   1  1

        1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5a 10a 4e 4f 5b 10b 2i
     2P 1a 1a 1a 1a 1a 1a 2c 2b 2g 1a 1a 1a 2c 5b  5b 2b 2g 5a  5a 1a
     3P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5b 10b 4e 4f 5a 10a 2i
     5P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 1a  2i 4e 4f 1a  2i 2i
     7P 1a 2a 2b 2c 2d 2e 4a 4b 4c 2f 2g 2h 4d 5b 10b 4e 4f 5a 10a 2i

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      2 -2  2  2 -2  .  .  .  . -2  2  .  .  A  -A  .  . *A -*A -2
X.6      2 -2  2  2 -2  .  .  .  . -2  2  .  . *A -*A  .  .  A  -A -2
X.7      2  2  2  2  2  .  .  .  .  2  2  .  .  A   A  .  . *A  *A  2
X.8      2  2  2  2  2  .  .  .  .  2  2  .  . *A  *A  .  .  A   A  2
X.9      5 -3  1  1  1 -1  1  1 -1  1 -3 -1  1  .   .  1 -1  .   .  5
X.10     5 -3  1  1  1  1 -1 -1  1  1 -3  1 -1  .   . -1  1  .   .  5
X.11     5  3  1  1 -1 -1 -1 -1 -1 -1 -3  1  1  .   .  1  1  .   . -5
X.12     5  3  1  1 -1  1  1  1  1 -1 -3 -1 -1  .   . -1 -1  .   . -5
X.13     5 -1 -3  1  3 -1 -1  1  1 -1  1  1  1  .   . -1 -1  .   . -5
X.14     5 -1 -3  1  3  1  1 -1 -1 -1  1 -1 -1  .   .  1  1  .   . -5
X.15     5 -1  1 -3 -1 -1  1 -1  1  3  1  1 -1  .   .  1 -1  .   . -5
X.16     5 -1  1 -3 -1  1 -1  1 -1  3  1 -1  1  .   . -1  1  .   . -5
X.17     5  1 -3  1 -3 -1  1 -1  1  1  1 -1  1  .   . -1  1  .   .  5
X.18     5  1 -3  1 -3  1 -1  1 -1  1  1  1 -1  .   .  1 -1  .   .  5
X.19     5  1  1 -3  1 -1 -1  1  1 -3  1 -1 -1  .   .  1  1  .   .  5
X.20     5  1  1 -3  1  1  1 -1 -1 -3  1  1  1  .   . -1 -1  .   .  5

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5