Properties

Label 40T7
Order \(40\)
n \(40\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_{10}$

Related objects

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $7$
Group :  $C_2^2\times C_{10}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,39)(38,40), (1,33,28,20,10,3,36,25,17,11)(2,34,27,19,9,4,35,26,18,12)(5,39,31,23,15,8,38,29,22,14)(6,40,32,24,16,7,37,30,21,13), (1,13,28,40,10,24,36,7,17,30)(2,14,27,39,9,23,35,8,18,29)(3,16,25,37,11,21,33,6,20,32)(4,15,26,38,12,22,34,5,19,31)
$|\Aut(F/K)|$:  $40$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
5:  $C_5$
8:  $C_2^3$
10:  $C_{10}$ x 7

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 5: $C_5$

Degree 8: $C_2^3$

Degree 10: $C_{10}$ x 7

Degree 20: 20T3 x 7

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, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,39)(38,40)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 5,10,15,17,22,28,31,36,38)( 2, 6, 9,16,18,21,27,32,35,37)( 3, 8,11,14,20, 23,25,29,33,39)( 4, 7,12,13,19,24,26,30,34,40)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 6,10,16,17,21,28,32,36,37)( 2, 5, 9,15,18,22,27,31,35,38)( 3, 7,11,13,20, 24,25,30,33,40)( 4, 8,12,14,19,23,26,29,34,39)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 7,10,13,17,24,28,30,36,40)( 2, 8, 9,14,18,23,27,29,35,39)( 3, 6,11,16,20, 21,25,32,33,37)( 4, 5,12,15,19,22,26,31,34,38)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 8,10,14,17,23,28,29,36,39)( 2, 7, 9,13,18,24,27,30,35,40)( 3, 5,11,15,20, 22,25,31,33,38)( 4, 6,12,16,19,21,26,32,34,37)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 9,17,27,36, 2,10,18,28,35)( 3,12,20,26,33, 4,11,19,25,34)( 5,16,22,32,38, 6,15,21,31,37)( 7,14,24,29,40, 8,13,23,30,39)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,10,17,28,36)( 2, 9,18,27,35)( 3,11,20,25,33)( 4,12,19,26,34) ( 5,15,22,31,38)( 6,16,21,32,37)( 7,13,24,30,40)( 8,14,23,29,39)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,11,17,25,36, 3,10,20,28,33)( 2,12,18,26,35, 4, 9,19,27,34)( 5,14,22,29,38, 8,15,23,31,39)( 6,13,21,30,37, 7,16,24,32,40)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,12,17,26,36, 4,10,19,28,34)( 2,11,18,25,35, 3, 9,20,27,33)( 5,13,22,30,38, 7,15,24,31,40)( 6,14,21,29,37, 8,16,23,32,39)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,13,28,40,10,24,36, 7,17,30)( 2,14,27,39, 9,23,35, 8,18,29)( 3,16,25,37,11, 21,33, 6,20,32)( 4,15,26,38,12,22,34, 5,19,31)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,14,28,39,10,23,36, 8,17,29)( 2,13,27,40, 9,24,35, 7,18,30)( 3,15,25,38,11, 22,33, 5,20,31)( 4,16,26,37,12,21,34, 6,19,32)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,15,28,38,10,22,36, 5,17,31)( 2,16,27,37, 9,21,35, 6,18,32)( 3,14,25,39,11, 23,33, 8,20,29)( 4,13,26,40,12,24,34, 7,19,30)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,16,28,37,10,21,36, 6,17,32)( 2,15,27,38, 9,22,35, 5,18,31)( 3,13,25,40,11, 24,33, 7,20,30)( 4,14,26,39,12,23,34, 8,19,29)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,17,36,10,28)( 2,18,35, 9,27)( 3,20,33,11,25)( 4,19,34,12,26) ( 5,22,38,15,31)( 6,21,37,16,32)( 7,24,40,13,30)( 8,23,39,14,29)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,18,36, 9,28, 2,17,35,10,27)( 3,19,33,12,25, 4,20,34,11,26)( 5,21,38,16,31, 6,22,37,15,32)( 7,23,40,14,30, 8,24,39,13,29)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,19,36,12,28, 4,17,34,10,26)( 2,20,35,11,27, 3,18,33, 9,25)( 5,24,38,13,31, 7,22,40,15,30)( 6,23,37,14,32, 8,21,39,16,29)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,20,36,11,28, 3,17,33,10,25)( 2,19,35,12,27, 4,18,34, 9,26)( 5,23,38,14,31, 8,22,39,15,29)( 6,24,37,13,32, 7,21,40,16,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,27)( 6,28)( 7,25)( 8,26)( 9,31)(10,32)(11,30) (12,29)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,28)( 6,27)( 7,26)( 8,25)( 9,32)(10,31)(11,29) (12,30)(13,34)(14,33)(15,36)(16,35)(17,38)(18,37)(19,40)(20,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,25)( 6,26)( 7,27)( 8,28)( 9,30)(10,29)(11,31) (12,32)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,24)( 2,23)( 3,21)( 4,22)( 5,26)( 6,25)( 7,28)( 8,27)( 9,29)(10,30)(11,32) (12,31)(13,36)(14,35)(15,34)(16,33)(17,40)(18,39)(19,38)(20,37)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,25,10,33,17, 3,28,11,36,20)( 2,26, 9,34,18, 4,27,12,35,19)( 5,29,15,39,22, 8,31,14,38,23)( 6,30,16,40,21, 7,32,13,37,24)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,26,10,34,17, 4,28,12,36,19)( 2,25, 9,33,18, 3,27,11,35,20)( 5,30,15,40,22, 7,31,13,38,24)( 6,29,16,39,21, 8,32,14,37,23)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,27,10,35,17, 2,28, 9,36,18)( 3,26,11,34,20, 4,25,12,33,19)( 5,32,15,37,22, 6,31,16,38,21)( 7,29,13,39,24, 8,30,14,40,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,28,10,36,17)( 2,27, 9,35,18)( 3,25,11,33,20)( 4,26,12,34,19) ( 5,31,15,38,22)( 6,32,16,37,21)( 7,30,13,40,24)( 8,29,14,39,23)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,29,17, 8,36,23,10,39,28,14)( 2,30,18, 7,35,24, 9,40,27,13)( 3,31,20, 5,33, 22,11,38,25,15)( 4,32,19, 6,34,21,12,37,26,16)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,30,17, 7,36,24,10,40,28,13)( 2,29,18, 8,35,23, 9,39,27,14)( 3,32,20, 6,33, 21,11,37,25,16)( 4,31,19, 5,34,22,12,38,26,15)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,31,17, 5,36,22,10,38,28,15)( 2,32,18, 6,35,21, 9,37,27,16)( 3,29,20, 8,33, 23,11,39,25,14)( 4,30,19, 7,34,24,12,40,26,13)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,32,17, 6,36,21,10,37,28,16)( 2,31,18, 5,35,22, 9,38,27,15)( 3,30,20, 7,33, 24,11,40,25,13)( 4,29,19, 8,34,23,12,39,26,14)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,33,28,20,10, 3,36,25,17,11)( 2,34,27,19, 9, 4,35,26,18,12)( 5,39,31,23,15, 8,38,29,22,14)( 6,40,32,24,16, 7,37,30,21,13)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,34,28,19,10, 4,36,26,17,12)( 2,33,27,20, 9, 3,35,25,18,11)( 5,40,31,24,15, 7,38,30,22,13)( 6,39,32,23,16, 8,37,29,21,14)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,35,28,18,10, 2,36,27,17, 9)( 3,34,25,19,11, 4,33,26,20,12)( 5,37,31,21,15, 6,38,32,22,16)( 7,39,30,23,13, 8,40,29,24,14)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,36,28,17,10)( 2,35,27,18, 9)( 3,33,25,20,11)( 4,34,26,19,12) ( 5,38,31,22,15)( 6,37,32,21,16)( 7,40,30,24,13)( 8,39,29,23,14)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,37,36,32,28,21,17,16,10, 6)( 2,38,35,31,27,22,18,15, 9, 5)( 3,40,33,30,25, 24,20,13,11, 7)( 4,39,34,29,26,23,19,14,12, 8)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,38,36,31,28,22,17,15,10, 5)( 2,37,35,32,27,21,18,16, 9, 6)( 3,39,33,29,25, 23,20,14,11, 8)( 4,40,34,30,26,24,19,13,12, 7)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,39,36,29,28,23,17,14,10, 8)( 2,40,35,30,27,24,18,13, 9, 7)( 3,38,33,31,25, 22,20,15,11, 5)( 4,37,34,32,26,21,19,16,12, 6)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,40,36,30,28,24,17,13,10, 7)( 2,39,35,29,27,23,18,14, 9, 8)( 3,37,33,32,25, 21,20,16,11, 6)( 4,38,34,31,26,22,19,15,12, 5)$

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [40, 14]
Character table: Data not available.