Properties

Label 35T10
Order \(210\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times F_7$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $10$
Group :  $C_5\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,28,25,2,29,21,3,30,22,4,26,23,5,27,24)(6,13,35,7,14,31,8,15,32,9,11,33,10,12,34)(16,18,20,17,19), (1,16,21,11,31,26)(2,17,22,12,32,27)(3,18,23,13,33,28)(4,19,24,14,34,29)(5,20,25,15,35,30)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
5:  $C_5$
6:  $C_6$
10:  $C_{10}$
15:  $C_{15}$
30:  $C_{30}$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Degree 7: $F_7$

Low degree siblings

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

Group invariants

Order:  $210=2 \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [210, 1]
Character table: Data not available.