↑

Properties

Label 35T18
Degree $35$
Order $840$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $D_7\times A_5$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(35, 18);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7\times A_5$
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:  (1,8,5,7,3,10)(2,6)(4,9)(11,33)(12,35,14,32,13,31)(15,34)(16,26,20,29,18,28)(17,27)(19,30)(22,24,25), (1,15,23,35,6,17,28,2,11,24,34,9,20,30,4,13,21,33,7,19,27)(3,14,25,31,8,16,29)(5,12,22,32,10,18,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$14$:  $D_{7}$
$60$:  $A_5$
$120$:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $A_5$

Degree 7: $D_{7}$

Low degree siblings

42T139

Siblings are shown 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 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,24, 7,28,13,35,20)( 2,21, 6,30,15,34,19)( 3,25, 8,29,14,31,16) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,29,16, 8,31,25,14) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $105$ $2$ $( 1,14)( 2,12)( 3,13)( 4,11)( 5,15)( 6,10)( 7, 8)(16,35)(17,33)(18,34)(19,32) (20,31)(21,26)(22,30)(23,27)(24,29)(25,28)$
$ 14, 14, 7 $ $30$ $14$ $( 1,29,20, 8,35,25,13, 3,28,16, 7,31,24,14)( 2,26,19,10,34,22,15, 5,30,18, 6, 32,21,12)( 4,27,17, 9,33,23,11)$
$ 14, 14, 7 $ $30$ $14$ $( 1,25, 7,29,13,31,20, 3,24, 8,28,14,35,16)( 2,22, 6,26,15,32,19, 5,21,10,30, 12,34,18)( 4,23, 9,27,11,33,17)$
$ 14, 14, 7 $ $30$ $14$ $( 1,31,28,25,20,14, 7, 3,35,29,24,16,13, 8)( 2,32,30,22,19,12, 6, 5,34,26,21, 18,15,10)( 4,33,27,23,17,11, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 3)( 2, 5)( 6,10)( 7, 8)(12,15)(13,14)(16,20)(18,19)(21,22)(24,25)(26,30) (28,29)(31,35)(32,34)$
$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $ $140$ $6$ $( 1,14, 2,13, 3,15)( 4,11)( 5,12)( 6, 7, 8)(16,34,20,31,19,35)(17,33)(18,32) (21,28,25,30,24,29)(22,26)(23,27)$
$ 21, 7, 7 $ $40$ $21$ $( 1,29,19, 7,31,21,13, 3,30,20, 8,34,24,14, 2,28,16, 6,35,25,15) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$
$ 21, 7, 7 $ $40$ $21$ $( 1,25, 6,28,14,34,20, 3,21, 7,29,15,35,16, 2,24, 8,30,13,31,19) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$
$ 21, 7, 7 $ $40$ $21$ $( 1,31,30,24,16,15, 7, 3,34,28,25,19,13, 8, 2,35,29,21,20,14, 6) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 3, 2)( 6, 7, 8)(13,14,15)(16,19,20)(21,24,25)(28,29,30)(31,34,35)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1,14, 2,12, 4,13, 3,15, 5,11)( 6,10, 9, 7, 8)(16,34,18,33,20,31,19,32,17,35) (21,26,23,28,25,30,22,27,24,29)$
$ 35 $ $24$ $35$ $( 1,29,19,10,33,24,14, 2,26,17, 7,31,21,12, 4,28,16, 6,32,23,13, 3,30,18, 9, 35,25,15, 5,27,20, 8,34,22,11)$
$ 35 $ $24$ $35$ $( 1,25, 6,26,11,35,16, 2,22, 9,28,14,34,18, 4,24, 8,30,12,33,20, 3,21,10,27, 13,31,19, 5,23, 7,29,15,32,17)$
$ 35 $ $24$ $35$ $( 1,31,30,22,17,13, 8, 2,32,27,24,16,15,10, 4,35,29,21,18,11, 7, 3,34,26,23, 20,14, 6, 5,33,28,25,19,12, 9)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 3, 2, 5, 4)( 6,10, 9, 7, 8)(11,13,14,15,12)(16,19,18,17,20) (21,22,23,24,25)(26,27,28,29,30)(31,34,32,33,35)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1,14, 2,11, 5,13, 3,15, 4,12)( 6, 9,10, 7, 8)(16,34,17,32,20,31,19,33,18,35) (21,27,22,28,25,30,23,26,24,29)$
$ 35 $ $24$ $35$ $( 1,29,19, 9,32,24,14, 2,27,18, 7,31,21,11, 5,28,16, 6,33,22,13, 3,30,17,10, 35,25,15, 4,26,20, 8,34,23,12)$
$ 35 $ $24$ $35$ $( 1,25, 6,27,12,35,16, 2,23,10,28,14,34,17, 5,24, 8,30,11,32,20, 3,21, 9,26, 13,31,19, 4,22, 7,29,15,33,18)$
$ 35 $ $24$ $35$ $( 1,31,30,23,18,13, 8, 2,33,26,24,16,15, 9, 5,35,29,21,17,12, 7, 3,34,27,22, 20,14, 6, 4,32,28,25,19,11,10)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 3, 2, 4, 5)( 6, 9,10, 7, 8)(11,12,13,14,15)(16,19,17,18,20) (21,23,22,24,25)(26,28,29,30,27)(31,34,33,32,35)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $840=2^{3} \cdot 3 \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  840.137
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 7A1 7A2 7A3 10A1 10A3 14A1 14A3 14A5 21A1 21A2 21A4 35A1 35A2 35A3 35A4 35A8 35A9
Size 1 7 15 105 20 12 12 140 2 2 2 84 84 30 30 30 40 40 40 24 24 24 24 24 24
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 7A2 7A3 7A1 5A1 5A2 7A3 7A2 7A1 21A2 21A4 21A1 35A3 35A1 35A9 35A2 35A4 35A8
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 7A3 7A1 7A2 10A3 10A1 14A3 14A5 14A1 7A1 7A2 7A3 35A8 35A9 35A4 35A3 35A1 35A2
5 P 1A 2A 2B 2C 3A 1A 1A 6A 7A2 7A3 7A1 2A 2A 14A5 14A1 14A3 21A2 21A4 21A1 7A2 7A3 7A1 7A1 7A2 7A3
7 P 1A 2A 2B 2C 3A 5A2 5A1 6A 1A 1A 1A 10A3 10A1 2B 2B 2B 3A 3A 3A 5A1 5A2 5A2 5A1 5A2 5A1
Type
840.137.1a R 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
840.137.1b R 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
840.137.2a1 R 2 0 2 0 2 2 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
840.137.2a2 R 2 0 2 0 2 2 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
840.137.2a3 R 2 0 2 0 2 2 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
840.137.3a1 R 3 3 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 3 3 3 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 −1 −1 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
840.137.3a2 R 3 3 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 3 3 3 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 −1 −1 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
840.137.3b1 R 3 −3 −1 1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 3 3 3 ζ5−2+ζ52 ζ5−1+ζ5 −1 −1 −1 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
840.137.3b2 R 3 −3 −1 1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 3 3 3 ζ5−1+ζ5 ζ5−2+ζ52 −1 −1 −1 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
840.137.4a R 4 4 0 0 1 −1 −1 1 4 4 4 −1 −1 0 0 0 1 1 1 −1 −1 −1 −1 −1 −1
840.137.4b R 4 −4 0 0 1 −1 −1 −1 4 4 4 1 1 0 0 0 1 1 1 −1 −1 −1 −1 −1 −1
840.137.5a R 5 5 1 1 −1 0 0 −1 5 5 5 0 0 1 1 1 −1 −1 −1 0 0 0 0 0 0
840.137.5b R 5 −5 1 −1 −1 0 0 1 5 5 5 0 0 1 1 1 −1 −1 −1 0 0 0 0 0 0
840.137.6a1 R 6 0 −2 0 0 −2ζ35−7−2ζ357 −2ζ35−14−2ζ3514 0 3ζ35−15+3ζ3515 3ζ35−5+3ζ355 3ζ35−10+3ζ3510 0 0 −ζ35−5−ζ355 −ζ35−15−ζ3515 −ζ35−10−ζ3510 0 0 0 ζ353−ζ354+ζ3510−ζ3511+ζ3517 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517
840.137.6a2 R 6 0 −2 0 0 −2ζ35−7−2ζ357 −2ζ35−14−2ζ3514 0 3ζ35−10+3ζ3510 3ζ35−15+3ζ3515 3ζ35−5+3ζ355 0 0 −ζ35−15−ζ3515 −ζ35−10−ζ3510 −ζ35−5−ζ355 0 0 0 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513 ζ353−ζ354+ζ3510−ζ3511+ζ3517
840.137.6a3 R 6 0 −2 0 0 −2ζ35−7−2ζ357 −2ζ35−14−2ζ3514 0 3ζ35−5+3ζ355 3ζ35−10+3ζ3510 3ζ35−15+3ζ3515 0 0 −ζ35−10−ζ3510 −ζ35−5−ζ355 −ζ35−15−ζ3515 0 0 0 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517 ζ353−ζ354+ζ3510−ζ3511+ζ3517 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516
840.137.6a4 R 6 0 −2 0 0 −2ζ35−14−2ζ3514 −2ζ35−7−2ζ357 0 3ζ35−15+3ζ3515 3ζ35−5+3ζ355 3ζ35−10+3ζ3510 0 0 −ζ35−5−ζ355 −ζ35−15−ζ3515 −ζ35−10−ζ3510 0 0 0 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513 ζ353−ζ354+ζ3510−ζ3511+ζ3517 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517
840.137.6a5 R 6 0 −2 0 0 −2ζ35−14−2ζ3514 −2ζ35−7−2ζ357 0 3ζ35−10+3ζ3510 3ζ35−15+3ζ3515 3ζ35−5+3ζ355 0 0 −ζ35−15−ζ3515 −ζ35−10−ζ3510 −ζ35−5−ζ355 0 0 0 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513 ζ353−ζ354+ζ3510−ζ3511+ζ3517 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517
840.137.6a6 R 6 0 −2 0 0 −2ζ35−14−2ζ3514 −2ζ35−7−2ζ357 0 3ζ35−5+3ζ355 3ζ35−10+3ζ3510 3ζ35−15+3ζ3515 0 0 −ζ35−10−ζ3510 −ζ35−5−ζ355 −ζ35−15−ζ3515 0 0 0 −ζ35−17−ζ35−12−ζ35−10−ζ35−5−ζ354+ζ357−ζ358−ζ359−ζ3511+ζ3512−ζ3513−ζ3516+ζ3517 −ζ35−17−ζ35−12−ζ35−7−ζ353+ζ355−ζ358−ζ359+ζ3512−ζ3513−ζ3516 ζ353−ζ354+ζ3510−ζ3511+ζ3517 ζ35−12+ζ35−10+ζ35−5+ζ352−ζ353+ζ354+ζ359+ζ3511+ζ3516−ζ3517 ζ35−17+ζ35−12−1+ζ354−ζ355−ζ357+ζ358+ζ359−ζ3510+ζ3511−ζ3512+ζ3513+ζ3516−ζ3517 ζ35−17+ζ35−7−ζ352+ζ353+ζ358−ζ3512+ζ3513
840.137.8a1 R 8 0 0 0 2 −2 −2 0 4ζ7−3+4ζ73 4ζ7−1+4ζ7 4ζ7−2+4ζ72 0 0 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
840.137.8a2 R 8 0 0 0 2 −2 −2 0 4ζ7−2+4ζ72 4ζ7−3+4ζ73 4ζ7−1+4ζ7 0 0 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
840.137.8a3 R 8 0 0 0 2 −2 −2 0 4ζ7−1+4ζ7 4ζ7−2+4ζ72 4ζ7−3+4ζ73 0 0 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7
840.137.10a1 R 10 0 2 0 −2 0 0 0 5ζ7−3+5ζ73 5ζ7−1+5ζ7 5ζ7−2+5ζ72 0 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 0 0 0 0 0 0
840.137.10a2 R 10 0 2 0 −2 0 0 0 5ζ7−2+5ζ72 5ζ7−3+5ζ73 5ζ7−1+5ζ7 0 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 0 0 0 0 0 0
840.137.10a3 R 10 0 2 0 −2 0 0 0 5ζ7−1+5ζ7 5ζ7−2+5ζ72 5ζ7−3+5ζ73 0 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 0 0 0 0 0 0

magma: CharacterTable(G);