↑

Properties

Label 20T63
Order \(240\)
n \(20\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_4\times A_5$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $63$
Group :  $C_4\times A_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,17,4,10,11,6,7,13,20,2,15,18,3,9,12,5,8,14,19), (1,6,18,9,13,2,5,17,10,14)(3,12,16,7,20,4,11,15,8,19)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
60:  $A_5$
120:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $A_5$

Degree 10: $A_5\times C_2$

Low degree siblings

24T574, 24T575, 40T174, 40T175

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 5,10,18)( 6, 9,17)( 7,15,19)( 8,16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $15$ $2$ $( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,18)(14,17)(15,19)(16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 6, 6, 2, 2, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4)( 5, 9,18, 6,10,17)( 7,16,19, 8,15,20)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,12)(13,17)(14,18)(15,20)(16,19)$
$ 4, 4, 4, 4, 4 $ $15$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,19,10,20)(11,14,12,13)(15,18,16,17)$
$ 20 $ $12$ $20$ $( 1, 3, 6, 7,10,11,14,15,18,20, 2, 4, 5, 8, 9,12,13,16,17,19)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 3, 6,12,13,16, 2, 4, 5,11,14,15)( 7,18, 8,17)( 9,19,10,20)$
$ 20 $ $12$ $20$ $( 1, 3, 6,19,18,11,14,15,10, 8, 2, 4, 5,20,17,12,13,16, 9, 7)$
$ 4, 4, 4, 4, 4 $ $15$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,20,10,19)(11,13,12,14)(15,17,16,18)$
$ 20 $ $12$ $20$ $( 1, 4, 6, 8,10,12,14,16,18,19, 2, 3, 5, 7, 9,11,13,15,17,20)$
$ 12, 4, 4 $ $20$ $12$ $( 1, 4, 6,11,13,15, 2, 3, 5,12,14,16)( 7,17, 8,18)( 9,20,10,19)$
$ 20 $ $12$ $20$ $( 1, 4, 6,20,18,12,14,16,10, 7, 2, 3, 5,19,17,11,13,15, 9, 8)$
$ 5, 5, 5, 5 $ $12$ $5$ $( 1, 5,10,13,18)( 2, 6, 9,14,17)( 3, 8,11,16,20)( 4, 7,12,15,19)$
$ 5, 5, 5, 5 $ $12$ $5$ $( 1, 5,10,18,13)( 2, 6, 9,17,14)( 3,11,16,20, 8)( 4,12,15,19, 7)$
$ 10, 10 $ $12$ $10$ $( 1, 6,10,14,18, 2, 5, 9,13,17)( 3, 7,11,15,20, 4, 8,12,16,19)$
$ 10, 10 $ $12$ $10$ $( 1, 6,10,17,13, 2, 5, 9,18,14)( 3,12,16,19, 8, 4,11,15,20, 7)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3,13, 4,14)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [240, 92]
Character table:    
      2  4  2  4  4  2  4  4   2   2   2  4   2   2   2  2  2   2   2  4  4
      3  1  1  .  1  1  .  .   .   1   .  .   .   1   .  .  .   .   .  1  1
      5  1  .  .  1  .  .  .   1   .   1  .   1   .   1  1  1   1   1  1  1

        1a 3a 2a 2b 6a 2c 4a 20a 12a 20b 4b 20c 12b 20d 5a 5b 10a 10b 4c 4d
     2P 1a 3a 1a 1a 3a 1a 2b 10a  6a 10b 2b 10a  6a 10b 5b 5a  5b  5a 2b 2b
     3P 1a 1a 2a 2b 2b 2c 4b 20d  4d 20c 4a 20b  4c 20a 5b 5a 10b 10a 4d 4c
     5P 1a 3a 2a 2b 6a 2c 4a  4c 12a  4c 4b  4d 12b  4d 1a 1a  2b  2b 4c 4d
     7P 1a 3a 2a 2b 6a 2c 4b 20d 12b 20c 4a 20b 12a 20a 5b 5a 10b 10a 4d 4c
    11P 1a 3a 2a 2b 6a 2c 4b 20c 12b 20d 4a 20a 12a 20b 5a 5b 10a 10b 4d 4c
    13P 1a 3a 2a 2b 6a 2c 4a 20b 12a 20a 4b 20d 12b 20c 5b 5a 10b 10a 4c 4d
    17P 1a 3a 2a 2b 6a 2c 4a 20b 12a 20a 4b 20d 12b 20c 5b 5a 10b 10a 4c 4d
    19P 1a 3a 2a 2b 6a 2c 4b 20c 12b 20d 4a 20a 12a 20b 5a 5b 10a 10b 4d 4c

X.1      1  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 -1 -1
X.3      1  1  1 -1 -1 -1  A   A   A   A -A  -A  -A  -A  1  1  -1  -1  A -A
X.4      1  1  1 -1 -1 -1 -A  -A  -A  -A  A   A   A   A  1  1  -1  -1 -A  A
X.5      3  . -1 -3  .  1  A   B   .   C -A  -B   .  -C *D  D -*D  -D  E -E
X.6      3  . -1 -3  .  1  A   C   .   B -A  -C   .  -B  D *D  -D -*D  E -E
X.7      3  . -1 -3  .  1 -A  -C   .  -B  A   C   .   B  D *D  -D -*D -E  E
X.8      3  . -1 -3  .  1 -A  -B   .  -C  A   B   .   C *D  D -*D  -D -E  E
X.9      3  . -1  3  . -1 -1   D   .  *D -1   D   .  *D *D  D  *D   D  3  3
X.10     3  . -1  3  . -1 -1  *D   .   D -1  *D   .   D  D *D   D  *D  3  3
X.11     3  . -1  3  . -1  1 -*D   .  -D  1 -*D   .  -D  D *D   D  *D -3 -3
X.12     3  . -1  3  . -1  1  -D   . -*D  1  -D   . -*D *D  D  *D   D -3 -3
X.13     4  1  .  4  1  .  .  -1   1  -1  .  -1   1  -1 -1 -1  -1  -1  4  4
X.14     4  1  .  4  1  .  .   1  -1   1  .   1  -1   1 -1 -1  -1  -1 -4 -4
X.15     4  1  . -4 -1  .  .  -A   A  -A  .   A  -A   A -1 -1   1   1  F -F
X.16     4  1  . -4 -1  .  .   A  -A   A  .  -A   A  -A -1 -1   1   1 -F  F
X.17     5 -1  1  5 -1  1  1   .  -1   .  1   .  -1   .  .  .   .   .  5  5
X.18     5 -1  1  5 -1  1 -1   .   1   . -1   .   1   .  .  .   .   . -5 -5
X.19     5 -1  1 -5  1 -1  A   .  -A   . -A   .   A   .  .  .   .   .  G -G
X.20     5 -1  1 -5  1 -1 -A   .   A   .  A   .  -A   .  .  .   .   . -G  G

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(20)-E(20)^9
C = -E(20)^13-E(20)^17
D = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
E = 3*E(4)
  = 3*Sqrt(-1) = 3i
F = -4*E(4)
  = -4*Sqrt(-1) = -4i
G = -5*E(4)
  = -5*Sqrt(-1) = -5i