Properties

Label 40T28
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{40}:C_2$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $28$
Group :  $C_{40}:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,27,37,12,23,34,6,20,31,4,13,25,39,10,21,35,8,18,30,2,15,28,38,11,24,33,5,19,32,3,14,26,40,9,22,36,7,17,29), (1,39,3,37,2,40,4,38)(5,36,8,33,6,35,7,34)(9,29,11,32,10,30,12,31)(13,25,16,27,14,26,15,28)(17,22,19,24,18,21,20,23)
$|\Aut(F/K)|$:  $4$

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:  $C_4\times C_2$
10:  $D_{5}$
16:  $C_8:C_2$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 8: $C_8:C_2$

Degree 10: $D_{10}$

Degree 20: 20T6

Low degree siblings

There are no siblings with degree $\leq 10$
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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 5,38)( 6,37)( 7,40)( 8,39)( 9,33)(10,34)(11,35)(12,36)(13,29)(14,30)(15,31) (16,32)(17,28)(18,27)(19,25)(20,26)(21,22)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,39, 6,40)( 7,38, 8,37)( 9,35,10,36)(11,34,12,33)(13,32,14,31) (15,29,16,30)(17,25,18,26)(19,27,20,28)(21,24,22,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,31,30,32)(33,36,34,35)(37,39,38,40)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,39,12,38,10,40,11,37)(13,33,15,36,14,34,16,35) (17,31,20,30,18,32,19,29)(21,28,24,26,22,27,23,25)$
$ 40 $ $2$ $40$ $( 1, 5,10,16,19,21,27,32,35,37, 3, 8,12,14,18,23,26,30,34,40, 2, 6, 9,15,20, 22,28,31,36,38, 4, 7,11,13,17,24,25,29,33,39)$
$ 40 $ $2$ $40$ $( 1, 6,10,15,19,22,27,31,35,38, 3, 7,12,13,18,24,26,29,34,39, 2, 5, 9,16,20, 21,28,32,36,37, 4, 8,11,14,17,23,25,30,33,40)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 7, 3, 5, 2, 8, 4, 6)( 9,38,11,39,10,37,12,40)(13,36,16,33,14,35,15,34) (17,30,19,31,18,29,20,32)(21,26,23,28,22,25,24,27)$
$ 40 $ $2$ $40$ $( 1, 7, 9,14,19,24,28,30,35,39, 4, 6,12,16,17,22,26,32,33,38, 2, 8,10,13,20, 23,27,29,36,40, 3, 5,11,15,18,21,25,31,34,37)$
$ 40 $ $2$ $40$ $( 1, 8, 9,13,19,23,28,29,35,40, 4, 5,12,15,17,21,26,31,33,37, 2, 7,10,14,20, 24,27,30,36,39, 3, 6,11,16,18,22,25,32,34,38)$
$ 20, 20 $ $2$ $20$ $( 1, 9,19,28,35, 4,12,17,26,33, 2,10,20,27,36, 3,11,18,25,34)( 5,15,21,31,37, 7,14,24,30,39, 6,16,22,32,38, 8,13,23,29,40)$
$ 20, 20 $ $2$ $20$ $( 1,10,19,27,35, 3,12,18,26,34, 2, 9,20,28,36, 4,11,17,25,33)( 5,16,21,32,37, 8,14,23,30,40, 6,15,22,31,38, 7,13,24,29,39)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,11,20,26,35)( 2,12,19,25,36)( 3,10,17,28,34)( 4, 9,18,27,33) ( 5,13,22,30,37)( 6,14,21,29,38)( 7,15,23,32,39)( 8,16,24,31,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,12,20,25,35, 2,11,19,26,36)( 3, 9,17,27,34, 4,10,18,28,33)( 5,14,22,29,37, 6,13,21,30,38)( 7,16,23,31,39, 8,15,24,32,40)$
$ 40 $ $2$ $40$ $( 1,13,28,40,12,21,33, 7,20,30, 3,16,25,38, 9,23,35, 5,17,31, 2,14,27,39,11, 22,34, 8,19,29, 4,15,26,37,10,24,36, 6,18,32)$
$ 40 $ $2$ $40$ $( 1,14,28,39,12,22,33, 8,20,29, 3,15,25,37, 9,24,35, 6,17,32, 2,13,27,40,11, 21,34, 7,19,30, 4,16,26,38,10,23,36, 5,18,31)$
$ 40 $ $2$ $40$ $( 1,15,27,38,12,24,34, 5,20,32, 4,14,25,40,10,22,35, 7,18,29, 2,16,28,37,11, 23,33, 6,19,31, 3,13,26,39, 9,21,36, 8,17,30)$
$ 40 $ $2$ $40$ $( 1,16,27,37,12,23,34, 6,20,31, 4,13,25,39,10,21,35, 8,18,30, 2,15,28,38,11, 24,33, 5,19,32, 3,14,26,40, 9,22,36, 7,17,29)$
$ 20, 20 $ $2$ $20$ $( 1,17,36, 9,26, 3,19,33,11,28, 2,18,35,10,25, 4,20,34,12,27)( 5,24,38,15,30, 8,21,39,13,31, 6,23,37,16,29, 7,22,40,14,32)$
$ 20, 20 $ $2$ $20$ $( 1,18,36,10,26, 4,19,34,11,27, 2,17,35, 9,25, 3,20,33,12,28)( 5,23,38,16,30, 7,21,40,13,32, 6,24,37,15,29, 8,22,39,14,31)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,35,12,26, 2,20,36,11,25)( 3,18,34, 9,28, 4,17,33,10,27)( 5,21,37,14,30, 6,22,38,13,29)( 7,24,39,16,32, 8,23,40,15,31)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,20,35,11,26)( 2,19,36,12,25)( 3,17,34,10,28)( 4,18,33, 9,27) ( 5,22,37,13,30)( 6,21,38,14,29)( 7,23,39,15,32)( 8,24,40,16,31)$
$ 8, 8, 8, 8, 8 $ $2$ $8$ $( 1,21, 3,23, 2,22, 4,24)( 5,27, 8,26, 6,28, 7,25)( 9,31,11,29,10,32,12,30) (13,33,16,35,14,34,15,36)(17,39,19,37,18,40,20,38)$
$ 8, 8, 8, 8, 8 $ $2$ $8$ $( 1,23, 4,21, 2,24, 3,22)( 5,26, 7,27, 6,25, 8,28)( 9,29,12,31,10,30,11,32) (13,35,15,33,14,36,16,34)(17,37,20,39,18,38,19,40)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 5]
Character table: Data not available.