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Properties

Label 20T196
Order \(1280\)
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 $t$ :  $196$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,8,13,11,15,17,4)(2,6,7,14,12,16,18,3)(9,19)(10,20), (1,17,11,8)(2,18,12,7)(3,16,14,6)(4,15,13,5)(9,20)(10,19), (1,2)(3,19)(4,20)(5,17,15,8)(6,18,16,7)(9,13)(10,14)(11,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $C_4\times C_2$ x 6, $C_2^3$
16:  $C_4\times C_2^2$
20:  $F_5$
40:  $F_{5}\times C_2$ x 3
80:  20T16
320:  $(C_2^4 : C_5):C_4$
640:  $((C_2^4 : C_5):C_4)\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$, $((C_2^4 : C_5):C_4)\times C_2$ x 2

Low degree siblings

20T196 x 11, 40T1028 x 6, 40T1032 x 6, 40T1048 x 3, 40T1049 x 3, 40T1052 x 3, 40T1056 x 3, 40T1058 x 4, 40T1059 x 4, 40T1146 x 6, 40T1150 x 6, 40T1152 x 6, 40T1153 x 3, 40T1155 x 3

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, 1, 1, 1, 1 $ $5$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,15)( 6,16)( 7,18)( 8,17)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 3,14)( 4,13)( 9,20)(10,19)$
$ 5, 5, 5, 5 $ $64$ $5$ $( 1,16,20,14,17)( 2,15,19,13,18)( 3, 8,11, 6, 9)( 4, 7,12, 5,10)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $20$ $4$ $( 3, 9,14,20)( 4,10,13,19)( 5, 7,15,18)( 6, 8,16,17)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,18,15, 7)( 6,17,16, 8)(13,19)(14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $20$ $2$ $( 3,20)( 4,19)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 5, 8,13,11,15,17, 4)( 2, 6, 7,14,12,16,18, 3)( 9,19)(10,20)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 5, 8, 4)( 2, 6, 7, 3)( 9,10)(11,15,17,13)(12,16,18,14)(19,20)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 7,20,13,11,18, 9, 4)( 2, 8,19,14,12,17,10, 3)( 5,16)( 6,15)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1,18, 9,13)( 2,17,10,14)( 3,12, 8,19)( 4,11, 7,20)( 5, 6)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1,11)( 2,12)( 5,15)( 6,16)( 7,18)( 8,17)$
$ 10, 10 $ $64$ $10$ $( 1,16,20, 3, 8,11, 6, 9,14,17)( 2,15,19, 4, 7,12, 5,10,13,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1, 1, 1 $ $40$ $4$ $( 3, 9)( 4,10)( 5, 7,15,18)( 6, 8,16,17)(13,19)(14,20)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1,11)( 2,12)( 3, 9,14,20)( 4,10,13,19)( 5,18,15, 7)( 6,17,16, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1,11)( 2,12)( 3,20)( 4,19)( 5, 7)( 6, 8)( 9,14)(10,13)(15,18)(16,17)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1, 5, 8,13,11,15,17, 4)( 2, 6, 7,14,12,16,18, 3)( 9,10)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 5, 8, 4)( 2, 6, 7, 3)( 9,19)(10,20)(11,15,17,13)(12,16,18,14)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1, 7,20, 4)( 2, 8,19, 3)( 5,16)( 6,15)( 9,13,11,18)(10,14,12,17)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1,18, 9, 4,11, 7,20,13)( 2,17,10, 3,12, 8,19,14)( 5, 6)(15,16)$
$ 8, 8, 1, 1, 1, 1 $ $40$ $8$ $( 3,16,20, 8,14, 6, 9,17)( 4,15,19, 7,13, 5,10,18)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1,11)( 2,12)( 3, 6, 9, 8)( 4, 5,10, 7)(13,15,19,18)(14,16,20,17)$
$ 8, 8, 1, 1, 1, 1 $ $40$ $8$ $( 3,17, 9, 6,14, 8,20,16)( 4,18,10, 5,13, 7,19,15)$
$ 4, 4, 4, 4, 2, 2 $ $40$ $4$ $( 1,11)( 2,12)( 3, 8,20, 6)( 4, 7,19, 5)( 9,16,14,17)(10,15,13,18)$
$ 10, 10 $ $64$ $10$ $( 1, 5,20,13, 8,12,16,10, 3,18)( 2, 6,19,14, 7,11,15, 9, 4,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,16)( 6,15)( 7, 8)( 9,10)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,17)( 8,18)( 9,10)(11,12)(13,14)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 7,11,18)( 2, 8,12,17)( 3, 5)( 4, 6)( 9,10)(13,16)(14,15)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 7,11,18)( 2, 8,12,17)( 3, 5,14,15)( 4, 6,13,16)( 9,19)(10,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)( 9,19)(10,20)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $40$ $4$ $( 3,16, 9,17)( 4,15,10,18)( 5,19, 7,13)( 6,20, 8,14)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1,11)( 2,12)( 3, 6,20,17,14,16, 9, 8)( 4, 5,19,18,13,15,10, 7)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $40$ $4$ $( 3,17, 9, 6)( 4,18,10, 5)( 7,19,15,13)( 8,20,16,14)$
$ 8, 8, 2, 2 $ $40$ $8$ $( 1,11)( 2,12)( 3, 8,20, 6,14,17, 9,16)( 4, 7,19, 5,13,18,10,15)$
$ 10, 10 $ $64$ $10$ $( 1, 5, 9, 4,17, 2, 6,10, 3,18)( 7,11,15,20,13, 8,12,16,19,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,13)( 4,14)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 7,11,18)( 2, 8,12,17)( 3, 5,14,15)( 4, 6,13,16)( 9,10)(19,20)$
$ 4, 4, 2, 2, 2, 2, 2, 2 $ $40$ $4$ $( 1, 7,11,18)( 2, 8,12,17)( 3, 5)( 4, 6)( 9,19)(10,20)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1,18)( 2,17)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)( 9,10)(19,20)$

Group invariants

Order:  $1280=2^{8} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1280, 1116439]
Character table: Data not available.