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Properties

Label 20T262
Order \(2560\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $262$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16)(2,15)(3,17,4,18)(5,20)(6,19)(7,11,8,12)(9,14)(10,13), (1,10,7,6,4)(2,9,8,5,3)(11,20,18,15,14)(12,19,17,16,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_{10}$

Low degree siblings

20T256 x 24, 20T262 x 23, 32T205514 x 8, 40T1883 x 48, 40T1975 x 24, 40T2019 x 12, 40T2128 x 48, 40T2166 x 24, 40T2206 x 24, 40T2209 x 48, 40T2210 x 48, 40T2211 x 48, 40T2212 x 24, 40T2240 x 12, 40T2291 x 8, 40T2292 x 24, 40T2293 x 48, 40T2294 x 48, 40T2295 x 48, 40T2296 x 48, 40T2297 x 96, 40T2298 x 96, 40T2299 x 96, 40T2300 x 96, 40T2301 x 96, 40T2302 x 96, 40T2303 x 96

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 4)( 7, 8)(11,12)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,16)( 2,15)( 3,17, 4,18)( 5,20)( 6,19)( 7,11, 8,12)( 9,14)(10,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(11,12)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 7, 8)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7, 8)( 9,10)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 9,10)(11,12)(13,14)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 5, 6)( 7, 8)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 5, 6)(11,12)(15,16)$
$ 4, 4, 4, 4, 2, 2 $ $80$ $4$ $( 1,16, 2,15)( 3,17)( 4,18)( 5,20, 6,19)( 7,11, 8,12)( 9,14,10,13)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 7, 8)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 5, 6)(11,12)(13,14)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 5, 6)( 9,10)(11,12)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 9,10)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $80$ $4$ $( 1,16)( 2,15)( 3,17, 4,18)( 5,20, 6,19)( 7,11)( 8,12)( 9,14)(10,13)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $(13,14)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)(17,18)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 5, 6)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 7, 8)( 9,10)(11,12)(19,20)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3, 4)( 9,10)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,16)( 2,15)( 3,17)( 4,18)( 5,20)( 6,19)( 7,11)( 8,12)( 9,14)(10,13)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1,10, 7, 6, 4)( 2, 9, 8, 5, 3)(11,20,18,15,14)(12,19,17,16,13)$
$ 10, 10 $ $256$ $10$ $( 1,13, 7,19, 4,16,10,11, 5,17)( 2,14, 8,20, 3,15, 9,12, 6,18)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 7, 4,10, 6)( 2, 8, 3, 9, 5)(11,18,14,20,15)(12,17,13,19,16)$
$ 10, 10 $ $256$ $10$ $( 1,11, 3,14, 5,15, 8,17,10,19)( 2,12, 4,13, 6,16, 7,18, 9,20)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,14,15,18,20)(12,13,16,17,19)$
$ 10, 10 $ $256$ $10$ $( 1,18, 5,12,10,16, 4,19, 7,13)( 2,17, 6,11, 9,15, 3,20, 8,14)$
$ 5, 5, 5, 5 $ $256$ $5$ $( 1, 6,10, 4, 7)( 2, 5, 9, 3, 8)(11,15,20,14,18)(12,16,19,13,17)$
$ 10, 10 $ $256$ $10$ $( 1,19,10,18, 8,15, 5,14, 3,12)( 2,20, 9,17, 7,16, 6,13, 4,11)$

Group invariants

Order:  $2560=2^{9} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.