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Properties

Label 20T74
Order \(320\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_2^4:C_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $74$
Group :  $C_2^2\times C_2^4:C_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,13,10,5,2,18,14,9,6)(3,20,15,11,7,4,19,16,12,8), (3,4)(5,16)(6,15)(7,17)(8,18)(9,20)(10,19)(13,14)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
10:  $C_{10}$ x 3
20:  20T3
80:  $C_2^4 : C_5$
160:  $C_2 \times (C_2^4 : C_5)$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_2 \times (C_2^4 : C_5)$ x 3

Low degree siblings

20T72 x 6, 20T74 x 35, 20T86 x 9, 40T202 x 3, 40T274 x 72, 40T275 x 36, 40T276 x 36, 40T277 x 18, 40T278 x 18, 40T292 x 9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1637]
Character table: Data not available.