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Properties

Label 35T6
35T6 1 5 1->5 9 1->9 2 8 2->8 3 4 3->4 7 3->7 4->1 6 4->6 5->3 10 5->10 6->10 14 6->14 13 7->13 8->9 12 8->12 9->6 11 9->11 10->8 15 10->15 11->15 19 11->19 18 12->18 13->14 17 13->17 14->11 16 14->16 15->13 20 15->20 16->20 24 16->24 23 17->23 18->19 22 18->22 19->16 21 19->21 20->18 25 20->25 21->25 29 21->29 28 22->28 23->24 27 23->27 24->21 26 24->26 25->23 30 25->30 26->30 34 26->34 33 27->33 28->29 32 28->32 29->26 31 29->31 30->28 35 30->35 31->4 31->35 32->3 33->2 33->34 34->1 34->31 35->5 35->33
Degree $35$
Order $140$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7\times F_5$

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Copy content magma:G := TransitiveGroup(35, 6);
 

Group invariants

Abstract group:  $C_7\times F_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $140=2^{2} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $6$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $7$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,11,19,21,29,31,4,6,14,16,24,26,34)(2,8,12,18,22,28,32,3,7,13,17,23,27,33)(5,10,15,20,25,30,35)$, $(1,5,3,4)(6,10,8,9)(11,15,13,14)(16,20,18,19)(21,25,23,24)(26,30,28,29)(31,35,33,34)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$7$:  $C_7$
$14$:  $C_{14}$
$20$:  $F_5$
$28$:  $C_{28}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $C_7$

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 TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{14},1^{7}$ $5$ $2$ $14$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)$
4A1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)(32,33,35,34)$
4A-1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28)(32,34,35,33)$
5A $5^{7}$ $4$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
7A1 $7^{5}$ $1$ $7$ $30$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33)( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
7A-1 $7^{5}$ $1$ $7$ $30$ $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8)( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$
7A2 $7^{5}$ $1$ $7$ $30$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
7A-2 $7^{5}$ $1$ $7$ $30$ $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13)( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$
7A3 $7^{5}$ $1$ $7$ $30$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23)( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
7A-3 $7^{5}$ $1$ $7$ $30$ $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18)( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$
14A1 $14^{2},7$ $5$ $14$ $32$ $( 1,21, 6,26,11,31,16)( 2,25, 7,30,12,35,17, 5,22,10,27,15,32,20)( 3,24, 8,29,13,34,18, 4,23, 9,28,14,33,19)$
14A-1 $14^{2},7$ $5$ $14$ $32$ $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25)( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$
14A3 $14^{2},7$ $5$ $14$ $32$ $( 1,26,16, 6,31,21,11)( 2,30,17,10,32,25,12, 5,27,20, 7,35,22,15)( 3,29,18, 9,33,24,13, 4,28,19, 8,34,23,14)$
14A-3 $14^{2},7$ $5$ $14$ $32$ $( 1,11,21,31, 6,16,26)( 2,15,22,35, 7,20,27, 5,12,25,32,10,17,30)( 3,14,23,34, 8,19,28, 4,13,24,33, 9,18,29)$
14A5 $14^{2},7$ $5$ $14$ $32$ $( 1,31,26,21,16,11, 6)( 2,35,27,25,17,15, 7, 5,32,30,22,20,12,10)( 3,34,28,24,18,14, 8, 4,33,29,23,19,13, 9)$
14A-5 $14^{2},7$ $5$ $14$ $32$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35)( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
28A1 $28,7$ $5$ $28$ $33$ $( 1,11,21,31, 6,16,26)( 2,14,25,33, 7,19,30, 3,12,24,35, 8,17,29, 5,13,22,34,10,18,27, 4,15,23,32, 9,20,28)$
28A-1 $28,7$ $5$ $28$ $33$ $( 1,30,18, 9,31,25,13, 4,26,20, 8,34,21,15, 3,29,16,10,33,24,11, 5,28,19, 6,35,23,14)( 2,27,17, 7,32,22,12)$
28A3 $28,7$ $5$ $28$ $33$ $( 1,33,27,25,16,13, 7, 5,31,28,22,20,11, 8, 2,35,26,23,17,15, 6, 3,32,30,21,18,12,10)( 4,34,29,24,19,14, 9)$
28A-3 $28,7$ $5$ $28$ $33$ $( 1, 6,11,16,21,26,31)( 2, 9,15,18,22,29,35, 3, 7,14,20,23,27,34, 5, 8,12,19,25,28,32, 4,10,13,17,24,30,33)$
28A5 $28,7$ $5$ $28$ $33$ $( 1,19,33,15,26, 9,23, 5,16,34,13,30, 6,24, 3,20,31,14,28,10,21, 4,18,35,11,29, 8,25)( 2,17,32,12,27, 7,22)$
28A-5 $28,7$ $5$ $28$ $33$ $( 1,21, 6,26,11,31,16)( 2,23,10,29,12,33,20, 4,22, 8,30,14,32,18, 5,24, 7,28,15,34,17, 3,25, 9,27,13,35,19)$
28A9 $28,7$ $5$ $28$ $33$ $( 1,21, 6,26,11,31,16)( 2,24,10,28,12,34,20, 3,22, 9,30,13,32,19, 5,23, 7,29,15,33,17, 4,25, 8,27,14,35,18)$
28A-9 $28,7$ $5$ $28$ $33$ $( 1,19,35,12,26, 9,25, 2,16,34,15,27, 6,24, 5,17,31,14,30, 7,21, 4,20,32,11,29,10,22)( 3,18,33,13,28, 8,23)$
28A11 $28,7$ $5$ $28$ $33$ $( 1, 6,11,16,21,26,31)( 2, 8,15,19,22,28,35, 4, 7,13,20,24,27,33, 5, 9,12,18,25,29,32, 3,10,14,17,23,30,34)$
28A-11 $28,7$ $5$ $28$ $33$ $( 1,33,29,22,16,13, 9, 2,31,28,24,17,11, 8, 4,32,26,23,19,12, 6, 3,34,27,21,18,14, 7)( 5,35,30,25,20,15,10)$
28A13 $28,7$ $5$ $28$ $33$ $( 1,30,17, 8,31,25,12, 3,26,20, 7,33,21,15, 2,28,16,10,32,23,11, 5,27,18, 6,35,22,13)( 4,29,19, 9,34,24,14)$
28A-13 $28,7$ $5$ $28$ $33$ $( 1,11,21,31, 6,16,26)( 2,13,25,34, 7,18,30, 4,12,23,35, 9,17,28, 5,14,22,33,10,19,27, 3,15,24,32, 8,20,29)$
35A1 $35$ $4$ $35$ $34$ $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24,16,13,10, 2,34,26,23,20,12, 9)$
35A-1 $35$ $4$ $35$ $34$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19,21,28,35, 2, 9,11,18,25,27,34)$
35A2 $35$ $4$ $35$ $34$ $( 1,30,19, 8,32,21,15, 4,28,17, 6,35,24,13, 2,26,20, 9,33,22,11, 5,29,18, 7,31,25,14, 3,27,16,10,34,23,12)$
35A-2 $35$ $4$ $35$ $34$ $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$
35A3 $35$ $4$ $35$ $34$ $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30,11,32,18, 4,25, 6,27,13,34,20)$
35A-3 $35$ $4$ $35$ $34$ $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13,26, 9,22, 5,18,31,14,27,10,23)$

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed