Properties

Label 10T25
Order \(320\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_2^4 : C_5):C_4$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $25$
Group :  $(C_2^4 : C_5):C_4$
CHM label :  $1/2[2^{5}]F(5)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,9,3)(2,4,8,6)(5,10), (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:  $C_2$
4:  $C_4$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $F_5$

Low degree siblings

10T24, 16T711, 20T77, 20T78, 20T79, 20T80, 20T83, 20T88, 32T9312, 40T206, 40T207, 40T296, 40T297, 40T298, 40T299, 40T300, 40T301, 40T302, 40T303

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 $ $10$ $2$ $( 4, 9)( 5,10)$
$ 8, 1, 1 $ $40$ $8$ $( 2, 3, 5, 9, 7, 8,10, 4)$
$ 8, 1, 1 $ $40$ $8$ $( 2, 4, 5, 8, 7, 9,10, 3)$
$ 2, 2, 2, 2, 1, 1 $ $20$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)$
$ 4, 4, 1, 1 $ $20$ $4$ $( 2, 5, 7,10)( 3, 4, 8, 9)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 4, 2, 2, 2 $ $40$ $4$ $( 1, 2)( 3, 5, 8,10)( 4, 9)( 6, 7)$
$ 5, 5 $ $64$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
$ 4, 4, 2 $ $40$ $4$ $( 1, 2, 4, 3)( 5,10)( 6, 7, 9, 8)$
$ 4, 4, 2 $ $40$ $4$ $( 1, 2, 5, 4)( 3, 8)( 6, 7,10, 9)$

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1635]
Character table:   
      2  6  5  3  3  4  4  6  3  .  3  3
      5  1  .  .  .  .  .  .  .  1  .  .

        1a 2a 8a 8b 2b 4a 2c 4b 5a 4c 4d
     2P 1a 1a 4a 4a 1a 2c 1a 2a 5a 2b 2b
     3P 1a 2a 8b 8a 2b 4a 2c 4b 5a 4d 4c
     5P 1a 2a 8a 8b 2b 4a 2c 4b 1a 4c 4d
     7P 1a 2a 8b 8a 2b 4a 2c 4b 5a 4d 4c

X.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
X.3      1  1  A -A -1 -1  1 -1  1  A -A
X.4      1  1 -A  A -1 -1  1 -1  1 -A  A
X.5      4  4  .  .  .  .  4  . -1  .  .
X.6      5  1 -1 -1  1  1 -3 -1  .  1  1
X.7      5  1  1  1  1  1 -3 -1  . -1 -1
X.8      5  1  A -A -1 -1 -3  1  . -A  A
X.9      5  1 -A  A -1 -1 -3  1  .  A -A
X.10    10 -2  .  . -2  2  2  .  .  .  .
X.11    10 -2  .  .  2 -2  2  .  .  .  .

A = -E(4)
  = -Sqrt(-1) = -i