↑

Properties

Label 35T18
35T18 1 8 1->8 15 1->15 2 6 2->6 11 2->11 3 10 3->10 14 3->14 4 9 4->9 13 4->13 5 7 5->7 12 5->12 17 6->17 7->3 19 7->19 8->5 16 8->16 20 9->20 10->1 18 10->18 24 11->24 33 11->33 22 12->22 35 12->35 21 13->21 31 13->31 25 14->25 32 14->32 23 15->23 34 15->34 26 16->26 29 16->29 27 17->27 28 17->28 18->26 18->28 19->27 30 19->30 20->29 20->30 21->33 22->24 22->32 23->35 24->25 24->34 25->22 25->31 26->5 26->20 27->1 28->2 28->16 29->3 29->18 30->4 31->8 31->12 32->10 32->13 33->7 34->9 35->6 35->14
Degree $35$
Order $840$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $D_7\times A_5$

Related objects

Downloads

Learn more

Show commands: Magma

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

Group invariants

Abstract group:  $D_7\times A_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $840=2^{3} \cdot 3 \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{15},1^{5}$ $7$ $2$ $15$ $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25)(17,23)(18,22)(19,21)(20,24)$
2B $2^{14},1^{7}$ $15$ $2$ $14$ $( 2, 4)( 3, 5)( 6, 9)( 8,10)(11,15)(12,14)(16,18)(17,19)(21,23)(22,25)(26,29)(27,30)(31,32)(33,34)$
2C $2^{17},1$ $105$ $2$ $17$ $( 1, 2)( 4, 5)( 6,35)( 7,34)( 8,31)( 9,32)(10,33)(11,26)(12,27)(13,30)(14,29)(15,28)(16,25)(17,22)(18,23)(19,24)(20,21)$
3A $3^{7},1^{14}$ $20$ $3$ $14$ $( 1, 3, 4)( 7, 8, 9)(11,13,14)(16,17,20)(23,24,25)(27,28,29)(31,33,35)$
5A1 $5^{7}$ $12$ $5$ $28$ $( 1, 3, 2, 4, 5)( 6, 9,10, 7, 8)(11,12,13,14,15)(16,19,17,18,20)(21,23,22,24,25)(26,28,29,30,27)(31,34,33,32,35)$
5A2 $5^{7}$ $12$ $5$ $28$ $( 1, 2, 5, 3, 4)( 6,10, 8, 9, 7)(11,13,15,12,14)(16,17,20,19,18)(21,22,25,23,24)(26,29,27,28,30)(31,33,35,34,32)$
6A $6^{3},3,2^{6},1^{2}$ $140$ $6$ $23$ $( 1, 4, 3)( 6,34)( 7,33, 8,35, 9,31)(10,32)(11,29,13,27,14,28)(12,26)(15,30)(16,24,17,25,20,23)(18,22)(19,21)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1,20,35,13,28, 7,24)( 2,19,34,15,30, 6,21)( 3,16,31,14,29, 8,25)( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$
7A2 $7^{5}$ $2$ $7$ $30$ $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8)( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
7A3 $7^{5}$ $2$ $7$ $30$ $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
10A1 $10^{3},5$ $84$ $10$ $31$ $( 1, 8, 5, 9, 2, 7, 3,10, 4, 6)(11,34,13,31,12,33,15,35,14,32)(16,26,17,30,20,29,18,27,19,28)(21,24,25,22,23)$
10A3 $10^{3},5$ $84$ $10$ $31$ $( 1, 9, 3, 6, 5, 7, 4, 8, 2,10)(11,31,15,32,13,33,14,34,12,35)(16,30,18,28,17,29,19,26,20,27)(21,22,24,23,25)$
14A1 $14^{2},7$ $30$ $14$ $32$ $( 1,35,28,24,20,13, 7)( 2,33,30,23,19,11, 6, 4,34,27,21,17,15, 9)( 3,32,29,22,16,12, 8, 5,31,26,25,18,14,10)$
14A3 $14^{2},7$ $30$ $14$ $32$ $( 1,24, 7,28,13,35,20)( 2,23, 6,27,15,33,19, 4,21, 9,30,11,34,17)( 3,22, 8,26,14,32,16, 5,25,10,29,12,31,18)$
14A5 $14^{2},7$ $30$ $14$ $32$ $( 1,13,24,35, 7,20,28)( 2,11,21,33, 6,17,30, 4,15,23,34, 9,19,27)( 3,12,25,32, 8,18,29, 5,14,22,31,10,16,26)$
21A1 $21,7^{2}$ $40$ $21$ $32$ $( 1,10,15,20,22,30,35, 5, 6,13,18,21,28,32, 2, 7,12,19,24,26,34)( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)$
21A2 $21,7^{2}$ $40$ $21$ $32$ $( 1,15,22,35, 6,18,28, 2,12,24,34,10,20,30, 5,13,21,32, 7,19,26)( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)$
21A4 $21,7^{2}$ $40$ $21$ $32$ $( 1,22, 6,28,12,34,20, 5,21, 7,26,15,35,18, 2,24,10,30,13,32,19)( 3,25, 8,29,14,31,16)( 4,23, 9,27,11,33,17)$
35A1 $35$ $24$ $35$ $34$ $( 1,11,25,32, 6,20,27, 3,12,21,35, 9,16,26, 2,13,23,31,10,19,28, 4,14,22,34, 7,17,29, 5,15,24,33, 8,18,30)$
35A2 $35$ $24$ $35$ $34$ $( 1,21, 9,29,12,35,19, 4,25,10,28,15,33,16, 5,24, 6,27,14,32,20, 2,23, 8,26,13,34,17, 3,22, 7,30,11,31,18)$
35A3 $35$ $24$ $35$ $34$ $( 1,34,27,25,18,13, 6, 4,31,26,24,19,11, 8, 5,35,30,23,16,12, 7, 2,33,29,22,20,15, 9, 3,32,28,21,17,14,10)$
35A4 $35$ $24$ $35$ $34$ $( 1,33,29,22,19,13, 9, 3,32,30,24,17,14,10, 2,35,27,25,18,15, 7, 4,31,26,21,20,11, 8, 5,34,28,23,16,12, 6)$
35A8 $35$ $24$ $35$ $34$ $( 1,15,23,31,10,20,30, 4,14,22,35, 6,17,29, 5,13,21,33, 8,18,28, 2,11,25,32, 7,19,27, 3,12,24,34, 9,16,26)$
35A9 $35$ $24$ $35$ $34$ $( 1,23, 8,26,15,35,17, 3,22, 6,28,11,31,18, 2,24, 9,29,12,34,20, 4,25,10,30,13,33,16, 5,21, 7,27,14,32,19)$

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(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 35A2 35A4 35A1 35A8 35A9 35A3
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 7A3 7A1 7A2 10A3 10A1 14A3 14A5 14A1 7A1 7A2 7A3 35A3 35A1 35A9 35A2 35A4 35A8
5 P 1A 2A 2B 2C 3A 1A 1A 6A 7A2 7A3 7A1 2A 2A 14A5 14A1 14A3 21A2 21A4 21A1 7A1 7A2 7A3 7A3 7A1 7A2
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 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
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 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
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 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
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 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
840.137.6a2 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 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
840.137.6a3 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−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
840.137.6a4 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 −ζ35−17−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
840.137.6a5 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 ζ35−12+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
840.137.6a6 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+ζ35−7−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
840.137.8a1 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 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7
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 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
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 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
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

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed