Properties

Label 10T22
Order \(240\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $S_5\times C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $22$
Group :  $S_5\times C_2$
CHM label :  $S(5)[x]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,10)(5,7), (1,6)(2,7)(3,8)(4,9)(5,10), (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:  $C_2^2$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $S_5$

Low degree siblings

10T22, 12T123 x 2, 20T62 x 2, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 5, 9)$
$ 3, 3, 1, 1, 1, 1 $ $20$ $3$ $( 3, 5, 9)( 4, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 4)( 3, 5)( 7, 9)( 8,10)$
$ 4, 4, 1, 1 $ $30$ $4$ $( 2, 4, 8,10)( 3, 5, 7, 9)$
$ 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4, 5, 8, 9,10)( 6, 7)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 8)( 4, 9)( 5,10)( 6, 7)$
$ 4, 4, 2 $ $30$ $4$ $( 1, 2, 3, 4)( 5,10)( 6, 7, 8, 9)$
$ 10 $ $24$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2, 3, 6, 7, 8)( 4, 9)( 5,10)$
$ 3, 3, 2, 2 $ $20$ $6$ $( 1, 3)( 2, 4,10)( 5, 7, 9)( 6, 8)$
$ 5, 5 $ $24$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [240, 189]
Character table:   
      2  4  3  2  4  3  4  2  3  3   1  2  2  1  4
      3  1  1  1  .  .  .  1  1  .   .  1  1  .  1
      5  1  .  .  .  .  .  .  .  .   1  .  .  1  1

        1a 2a 3a 2b 4a 2c 6a 2d 4b 10a 6b 6c 5a 2e
     2P 1a 1a 3a 1a 2b 1a 3a 1a 2b  5a 3a 3a 5a 1a
     3P 1a 2a 1a 2b 4a 2c 2d 2d 4b 10a 2e 2a 5a 2e
     5P 1a 2a 3a 2b 4a 2c 6a 2d 4b  2e 6b 6c 1a 2e
     7P 1a 2a 3a 2b 4a 2c 6a 2d 4b 10a 6b 6c 5a 2e

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