Properties

Label 10T35
Order \(1440\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $(A_6 : C_2) : C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $35$
Group :  $(A_6 : C_2) : C_2$
CHM label :  $L(10).2^{2}=P|L(2,9)$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,7,3,4,2,5,6,8), (3,6)(4,7)(5,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

12T220, 20T201, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187

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, 5 $ $144$ $5$ $( 1, 7, 6, 4, 2)( 3, 9, 5, 8,10)$
$ 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 8)( 2, 5)( 3, 6)( 4, 9)( 7,10)$
$ 10 $ $144$ $10$ $( 1,10, 6, 9, 2, 8, 7, 3, 4, 5)$
$ 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 2, 7)( 3, 9)( 4, 6)( 5,10)$
$ 4, 4, 2 $ $90$ $4$ $( 1, 8)( 2, 3, 7, 9)( 4, 5, 6,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $30$ $2$ $( 2, 3)( 5,10)( 7, 9)$
$ 4, 4, 1, 1 $ $180$ $4$ $( 1, 6, 4, 7)( 3, 9,10, 8)$
$ 4, 4, 1, 1 $ $90$ $4$ $( 2, 3, 7, 9)( 4,10, 6, 5)$
$ 8, 2 $ $180$ $8$ $( 1, 8)( 2, 4, 3,10, 7, 6, 9, 5)$
$ 3, 3, 3, 1 $ $80$ $3$ $( 1, 6, 3)( 4,10, 7)( 5, 8, 9)$
$ 6, 3, 1 $ $240$ $6$ $( 1, 3, 6)( 4, 5,10, 8, 7, 9)$
$ 8, 1, 1 $ $180$ $8$ $( 1, 8, 4,10, 7, 6, 9, 2)$

Group invariants

Order:  $1440=2^{5} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1440, 5841]
Character table:   
      2  5  5  3  4  3  1   1  4  3  1  4  1  3
      3  2  .  .  .  .  .   .  .  .  2  1  1  .
      5  1  .  .  .  1  1   1  .  .  .  .  .  .

        1a 2a 4a 4b 2b 5a 10a 4c 8a 3a 2c 6a 8b
     2P 1a 1a 2a 2a 1a 5a  5a 2a 4c 3a 1a 3a 4c
     3P 1a 2a 4a 4b 2b 5a 10a 4c 8a 1a 2c 2c 8b
     5P 1a 2a 4a 4b 2b 1a  2b 4c 8a 3a 2c 6a 8b
     7P 1a 2a 4a 4b 2b 5a 10a 4c 8a 3a 2c 6a 8b

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      9  1 -1 -1  1 -1   1  1 -1  .  3  .  1
X.6      9  1 -1  1 -1 -1  -1  1  1  . -3  .  1
X.7      9  1  1 -1 -1 -1  -1  1  1  .  3  . -1
X.8      9  1  1  1  1 -1   1  1 -1  . -3  . -1
X.9     10  2  .  2  .  .   . -2  .  1  2 -1  .
X.10    10  2  . -2  .  .   . -2  .  1 -2  1  .
X.11    16  .  .  . -4  1   1  .  . -2  .  .  .
X.12    16  .  .  .  4  1  -1  .  . -2  .  .  .
X.13    20 -4  .  .  .  .   .  .  .  2  .  .  .