Properties

Label 35T14
Order \(420\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_5\times C_7:C_3$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $14$
Group :  $F_5\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,9,28,11,32,19,3,21,7,29,13,31,17,4,23,6,27,14,33,16,2,24,8,26,12,34,18)(5,25,10,30,15,35,20), (1,8,16,3,6,18)(2,7,17)(4,10,19,5,9,20)(11,28,21,13,26,23)(12,27,22)(14,30,24,15,29,25)(31,33)(34,35)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
20:  $F_5$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$
60:  $F_5\times C_3$
84:  28T13

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $C_7:C_3$

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

Group invariants

Order:  $420=2^{2} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [420, 14]
Character table: Data not available.