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Properties

Label 10T37
Order \(1920\)
n \(10\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $(C_2^4:A_5) : C_2$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $37$
Group :  $(C_2^4:A_5) : C_2$
CHM label :  $[2^{4}]S(5)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,10)(5,7), (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)
2:  $C_2$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $S_5$

Low degree siblings

10T38, 16T1328, 20T218, 20T219, 20T222, 20T223, 20T226, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678

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, 1, 1, 1, 1, 1, 1 $ $20$ $2$ $( 4,10)( 5, 9)$
$ 4, 2, 1, 1, 1, 1 $ $60$ $4$ $( 2, 7)( 4, 5, 9,10)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 1, 6)( 2, 7)( 4,10)( 5, 9)$
$ 4, 2, 2, 2 $ $20$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 4, 5, 9,10)$
$ 3, 3, 1, 1, 1, 1 $ $80$ $3$ $( 3, 9, 5)( 4,10, 8)$
$ 6, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 9,10, 8, 4, 5)$
$ 3, 3, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 9, 5)( 4,10, 8)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 2, 8)( 3, 7)( 4,10)( 5, 9)$
$ 4, 4, 1, 1 $ $60$ $4$ $( 2, 8, 7, 3)( 4, 5, 9,10)$
$ 4, 2, 2, 2 $ $120$ $4$ $( 1, 6)( 2, 8)( 3, 7)( 4, 5, 9,10)$
$ 4, 4, 1, 1 $ $240$ $4$ $( 2, 8, 4,10)( 3, 9, 5, 7)$
$ 8, 2 $ $240$ $8$ $( 1, 6)( 2, 8, 4, 5, 7, 3, 9,10)$
$ 3, 3, 2, 2 $ $160$ $6$ $( 1, 7)( 2, 6)( 3, 9, 5)( 4,10, 8)$
$ 6, 4 $ $160$ $12$ $( 1, 2, 6, 7)( 3, 9,10, 8, 4, 5)$
$ 5, 5 $ $384$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$

Group invariants

Order:  $1920=2^{7} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  7  6  7  5  5  5  5  3  2  3  5  5  4  3  3  2   2  .
      3  1  1  1  1  .  .  1  1  1  1  .  .  .  .  .  1   1  .
      5  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .   .  1

        1a 2a 2b 2c 4a 2d 4b 3a 6a 6b 2e 4c 4d 4e 8a 6c 12a 5a
     2P 1a 1a 1a 1a 2a 1a 2a 3a 3a 3a 1a 2b 2a 2e 4c 3a  6b 5a
     3P 1a 2a 2b 2c 4a 2d 4b 1a 2b 2a 2e 4c 4d 4e 8a 2c  4b 5a
     5P 1a 2a 2b 2c 4a 2d 4b 3a 6a 6b 2e 4c 4d 4e 8a 6c 12a 1a
     7P 1a 2a 2b 2c 4a 2d 4b 3a 6a 6b 2e 4c 4d 4e 8a 6c 12a 5a
    11P 1a 2a 2b 2c 4a 2d 4b 3a 6a 6b 2e 4c 4d 4e 8a 6c 12a 5a

X.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
X.3      4  4  4 -2 -2 -2 -2  1  1  1  .  .  .  .  .  1   1 -1
X.4      4  4  4  2  2  2  2  1  1  1  .  .  .  .  . -1  -1 -1
X.5      5  5  5 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1 -1  -1  .
X.6      5  5  5  1  1  1  1 -1 -1 -1  1  1  1 -1 -1  1   1  .
X.7      5  1 -3  3  1 -1 -3  2  . -2  1  1 -1  1 -1  .   .  .
X.8      5  1 -3 -3 -1  1  3  2  . -2  1  1 -1 -1  1  .   .  .
X.9      6  6  6  .  .  .  .  .  .  . -2 -2 -2  .  .  .   .  1
X.10    10 -2  2 -4  2  . -2  1 -1  1  2 -2  .  .  . -1   1  .
X.11    10 -2  2 -2  .  2 -4  1 -1  1 -2  2  .  .  .  1  -1  .
X.12    10 -2  2  4 -2  .  2  1 -1  1  2 -2  .  .  .  1  -1  .
X.13    10 -2  2  2  . -2  4  1 -1  1 -2  2  .  .  . -1   1  .
X.14    10  2 -6  .  .  .  . -2  .  2  2  2 -2  .  .  .   .  .
X.15    15  3 -9 -3 -1  1  3  .  .  . -1 -1  1  1 -1  .   .  .
X.16    15  3 -9  3  1 -1 -3  .  .  . -1 -1  1 -1  1  .   .  .
X.17    20 -4  4 -2  2 -2  2 -1  1 -1  .  .  .  .  .  1  -1  .
X.18    20 -4  4  2 -2  2 -2 -1  1 -1  .  .  .  .  . -1   1  .