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Properties

Label 35T4
Degree $35$
Order $70$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{35}$

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Show commands: Magma

magma: G := TransitiveGroup(35, 4);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{35}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,5)(3,4)(6,31)(7,35)(8,34)(9,33)(10,32)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22), (1,29)(2,28)(3,27)(4,26)(5,30)(6,24)(7,23)(8,22)(9,21)(10,25)(11,19)(12,18)(13,17)(14,16)(15,20)(31,34)(32,33)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$
$14$:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $D_{7}$

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $70=2 \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  70.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 5A1 5A2 7A1 7A2 7A3 35A1 35A2 35A3 35A4 35A6 35A8 35A9 35A11 35A12 35A13 35A16 35A17
Size 1 35 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 5A2 5A1 7A3 7A1 7A2 35A16 35A9 35A12 35A6 35A8 35A1 35A2 35A17 35A3 35A4 35A11 35A13
5 P 1A 2A 5A2 5A1 7A1 7A2 7A3 35A11 35A4 35A17 35A9 35A12 35A16 35A3 35A8 35A13 35A6 35A1 35A2
7 P 1A 2A 1A 1A 7A3 7A1 7A2 7A1 7A1 7A1 7A3 7A3 7A3 7A1 7A2 7A2 7A2 7A2 7A3
Type
70.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
70.3.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
70.3.2a1 R 2 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
70.3.2a2 R 2 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
70.3.2b1 R 2 0 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
70.3.2b2 R 2 0 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
70.3.2b3 R 2 0 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
70.3.2c1 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−17+ζ3517 ζ35−1+ζ35 ζ35−16+ζ3516 ζ35−2+ζ352 ζ35−3+ζ353 ζ35−4+ζ354 ζ35−13+ζ3513 ζ35−12+ζ3512 ζ35−6+ζ356 ζ35−11+ζ3511 ζ35−8+ζ358 ζ35−9+ζ359
70.3.2c2 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−3+ζ353 ζ35−6+ζ356 ζ35−9+ζ359 ζ35−12+ζ3512 ζ35−17+ζ3517 ζ35−11+ζ3511 ζ35−8+ζ358 ζ35−2+ζ352 ζ35−1+ζ35 ζ35−4+ζ354 ζ35−13+ζ3513 ζ35−16+ζ3516
70.3.2c3 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−12+ζ3512 ζ35−11+ζ3511 ζ35−1+ζ35 ζ35−13+ζ3513 ζ35−2+ζ352 ζ35−9+ζ359 ζ35−3+ζ353 ζ35−8+ζ358 ζ35−4+ζ354 ζ35−16+ζ3516 ζ35−17+ζ3517 ζ35−6+ζ356
70.3.2c4 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−2+ζ352 ζ35−4+ζ354 ζ35−6+ζ356 ζ35−8+ζ358 ζ35−12+ζ3512 ζ35−16+ζ3516 ζ35−17+ζ3517 ζ35−13+ζ3513 ζ35−11+ζ3511 ζ35−9+ζ359 ζ35−3+ζ353 ζ35−1+ζ35
70.3.2c5 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−13+ζ3513 ζ35−9+ζ359 ζ35−4+ζ354 ζ35−17+ζ3517 ζ35−8+ζ358 ζ35−1+ζ35 ζ35−12+ζ3512 ζ35−3+ζ353 ζ35−16+ζ3516 ζ35−6+ζ356 ζ35−2+ζ352 ζ35−11+ζ3511
70.3.2c6 R 2 0 ζ35−14+ζ3514 ζ35−7+ζ357 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−8+ζ358 ζ35−16+ζ3516 ζ35−11+ζ3511 ζ35−3+ζ353 ζ35−13+ζ3513 ζ35−6+ζ356 ζ35−2+ζ352 ζ35−17+ζ3517 ζ35−9+ζ359 ζ35−1+ζ35 ζ35−12+ζ3512 ζ35−4+ζ354
70.3.2c7 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−11+ζ3511 ζ35−13+ζ3513 ζ35−2+ζ352 ζ35−9+ζ359 ζ35−4+ζ354 ζ35−17+ζ3517 ζ35−6+ζ356 ζ35−16+ζ3516 ζ35−8+ζ358 ζ35−3+ζ353 ζ35−1+ζ35 ζ35−12+ζ3512
70.3.2c8 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−4+ζ354 ζ35−8+ζ358 ζ35−12+ζ3512 ζ35−16+ζ3516 ζ35−11+ζ3511 ζ35−3+ζ353 ζ35−1+ζ35 ζ35−9+ζ359 ζ35−13+ζ3513 ζ35−17+ζ3517 ζ35−6+ζ356 ζ35−2+ζ352
70.3.2c9 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−16+ζ3516 ζ35−3+ζ353 ζ35−13+ζ3513 ζ35−6+ζ356 ζ35−9+ζ359 ζ35−12+ζ3512 ζ35−4+ζ354 ζ35−1+ζ35 ζ35−17+ζ3517 ζ35−2+ζ352 ζ35−11+ζ3511 ζ35−8+ζ358
70.3.2c10 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−5+ζ355 ζ35−9+ζ359 ζ35−17+ζ3517 ζ35−8+ζ358 ζ35−1+ζ35 ζ35−16+ζ3516 ζ35−2+ζ352 ζ35−11+ζ3511 ζ35−6+ζ356 ζ35−3+ζ353 ζ35−12+ζ3512 ζ35−4+ζ354 ζ35−13+ζ3513
70.3.2c11 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−6+ζ356 ζ35−12+ζ3512 ζ35−17+ζ3517 ζ35−11+ζ3511 ζ35−1+ζ35 ζ35−13+ζ3513 ζ35−16+ζ3516 ζ35−4+ζ354 ζ35−2+ζ352 ζ35−8+ζ358 ζ35−9+ζ359 ζ35−3+ζ353
70.3.2c12 R 2 0 ζ35−7+ζ357 ζ35−14+ζ3514 ζ35−5+ζ355 ζ35−10+ζ3510 ζ35−15+ζ3515 ζ35−1+ζ35 ζ35−2+ζ352 ζ35−3+ζ353 ζ35−4+ζ354 ζ35−6+ζ356 ζ35−8+ζ358 ζ35−9+ζ359 ζ35−11+ζ3511 ζ35−12+ζ3512 ζ35−13+ζ3513 ζ35−16+ζ3516 ζ35−17+ζ3517

magma: CharacterTable(G);