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Properties

Label 20T33
Order \(120\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:S_4$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $33$
Group :  $C_5:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,17,5,13)(2,11,20,6,15,4,10,19,8,14,3,12,18,7,16), (1,7,4,6)(2,5,3,8)(9,19,12,18)(10,17,11,20)(13,15,16,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
10:  $D_{5}$
24:  $S_4$
30:  $D_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 5: $D_{5}$

Degree 10: None

Low degree siblings

30T19, 30T31, 40T63

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $30$ $2$ $( 3, 4)( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 4 $ $30$ $4$ $( 1, 2, 3, 4)( 5,18, 7,20)( 6,19, 8,17)( 9,14,11,16)(10,15,12,13)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$
$ 15, 5 $ $8$ $15$ $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$
$ 15, 5 $ $8$ $15$ $( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$
$ 10, 10 $ $6$ $10$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
$ 15, 5 $ $8$ $15$ $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$
$ 15, 5 $ $8$ $15$ $( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$
$ 10, 10 $ $6$ $10$ $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [120, 38]
Character table:    
      2  3  2  .  3  2  2   .   .   2  2   .   .   2
      3  1  .  1  .  .  1   1   1   .  1   1   1   .
      5  1  .  1  1  .  1   1   1   1  1   1   1   1

        1a 2a 3a 2b 4a 5a 15a 15b 10a 5b 15c 15d 10b
     2P 1a 1a 3a 1a 2b 5b 15d 15c  5b 5a 15a 15b  5a
     3P 1a 2a 1a 2b 4a 5b  5b  5b 10b 5a  5a  5a 10a
     5P 1a 2a 3a 2b 4a 1a  3a  3a  2b 1a  3a  3a  2b
     7P 1a 2a 3a 2b 4a 5b 15c 15d 10b 5a 15b 15a 10a
    11P 1a 2a 3a 2b 4a 5a 15b 15a 10a 5b 15d 15c 10b
    13P 1a 2a 3a 2b 4a 5b 15d 15c 10b 5a 15a 15b 10a

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      2  . -1  2  .  2  -1  -1   2  2  -1  -1   2
X.4      2  .  2  2  .  A   A   A   A *A  *A  *A  *A
X.5      2  .  2  2  . *A  *A  *A  *A  A   A   A   A
X.6      2  . -1  2  . *A   C   D  *A  A   F   E   A
X.7      2  . -1  2  . *A   D   C  *A  A   E   F   A
X.8      2  . -1  2  .  A   E   F   A *A   C   D  *A
X.9      2  . -1  2  .  A   F   E   A *A   D   C  *A
X.10     3 -1  . -1  1  3   .   .  -1  3   .   .  -1
X.11     3  1  . -1 -1  3   .   .  -1  3   .   .  -1
X.12     6  .  . -2  .  B   .   .  -A *B   .   . -*A
X.13     6  .  . -2  . *B   .   . -*A  B   .   .  -A

A = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5
B = 3*E(5)+3*E(5)^4
  = (-3+3*Sqrt(5))/2 = 3b5
C = E(15)^4+E(15)^11
D = E(15)+E(15)^14
E = E(15)^7+E(15)^8
F = E(15)^2+E(15)^13