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Properties

Label 10T34
Order \(960\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2^4 : A_5$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $34$
Group :  $C_2^4 : A_5$
CHM label :  $[2^{4}]A(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,4,10)(5,7,9), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
60:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $A_5$

Low degree siblings

16T1081, 20T172, 20T177, 30T214, 30T217, 40T888, 40T889, 40T932, 40T942, 40T944, 40T945

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 $ $10$ $2$ $( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 1, 2)( 3, 4)( 6, 7)( 8, 9)$
$ 4, 2, 2, 2 $ $120$ $4$ $( 1, 2)( 3, 9, 8, 4)( 5,10)( 6, 7)$
$ 4, 4, 1, 1 $ $60$ $4$ $( 1, 7, 6, 2)( 3, 9, 8, 4)$
$ 3, 3, 1, 1, 1, 1 $ $80$ $3$ $( 1, 2, 3)( 6, 7, 8)$
$ 3, 3, 2, 2 $ $80$ $6$ $( 1, 2, 3)( 4, 9)( 5,10)( 6, 7, 8)$
$ 6, 2, 1, 1 $ $80$ $6$ $( 1, 7, 8, 6, 2, 3)( 5,10)$
$ 6, 2, 1, 1 $ $80$ $6$ $( 1, 7, 8, 6, 2, 3)( 4, 9)$
$ 5, 5 $ $192$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
$ 5, 5 $ $192$ $5$ $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)$

Group invariants

Order:  $960=2^{6} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [960, 11358]
Character table:    
      2  6  5  6  4  3  4  2  2  2  2  .  .
      3  1  1  1  .  .  .  1  1  1  1  .  .
      5  1  .  .  .  .  .  .  .  .  .  1  1

        1a 2a 2b 2c 4a 4b 3a 6a 6b 6c 5a 5b
     2P 1a 1a 1a 1a 2a 2b 3a 3a 3a 3a 5b 5a
     3P 1a 2a 2b 2c 4a 4b 1a 2a 2b 2b 5b 5a
     5P 1a 2a 2b 2c 4a 4b 3a 6a 6c 6b 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1  1
X.2      3  3  3 -1 -1 -1  .  .  .  .  B *B
X.3      3  3  3 -1 -1 -1  .  .  .  . *B  B
X.4      4  4  4  .  .  .  1  1  1  1 -1 -1
X.5      5  5  5  1  1  1 -1 -1 -1 -1  .  .
X.6      5  1 -3  1 -1  1  2 -2  .  .  .  .
X.7      5  1 -3  1 -1  1 -1  1  A -A  .  .
X.8      5  1 -3  1 -1  1 -1  1 -A  A  .  .
X.9     10 -2  2 -2  .  2  1  1 -1 -1  .  .
X.10    10 -2  2  2  . -2  1  1 -1 -1  .  .
X.11    15  3 -9 -1  1 -1  .  .  .  .  .  .
X.12    20 -4  4  .  .  . -1 -1  1  1  .  .

A = -E(3)+E(3)^2
  = -Sqrt(-3) = -i3
B = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5