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Properties

Label 20T18
Order \(80\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4:F_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $18$
Group :  $C_4:F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15,9,3,17,12,6,20,14,8,2,16,10,4,18,11,5,19,13,7), (1,8,10,4)(2,7,9,3)(5,15)(6,16)(11,18,20,13)(12,17,19,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $C_4:C_4$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T18, 40T52, 40T54

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 31]
Character table:    
      2  4  3  3  4  4  4  3   2  3  3   2  2   2  3
      5  1  .  .  .  1  .  .   1  .  .   1  1   1  1

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

X.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
X.3      1 -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 -1
X.5      1  A -A -1  1 -1 -1   1  A -A   1  1   1  1
X.6      1 -A  A -1  1 -1 -1   1 -A  A   1  1   1  1
X.7      1  A -A -1  1 -1  1  -1 -A  A  -1  1   1 -1
X.8      1 -A  A -1  1 -1  1  -1  A -A  -1  1   1 -1
X.9      2  .  . -2 -2  2  .   .  .  .   .  2  -2  .
X.10     2  .  .  2 -2 -2  .   .  .  .   .  2  -2  .
X.11     4  .  .  .  4  .  .  -1  .  .  -1 -1  -1  4
X.12     4  .  .  .  4  .  .   1  .  .   1 -1  -1 -4
X.13     4  .  .  . -4  .  .   B  .  .  -B -1   1  .
X.14     4  .  .  . -4  .  .  -B  .  .   B -1   1  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(20)-E(20)^9+E(20)^13+E(20)^17
  = -Sqrt(-5) = -i5