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Properties

Label 35T10
35T10 1 16 1->16 28 1->28 2 17 2->17 29 2->29 3 18 3->18 30 3->30 4 19 4->19 26 4->26 5 20 5->20 27 5->27 6 13 6->13 7 14 7->14 8 15 8->15 9 11 9->11 10 12 10->12 31 11->31 33 11->33 32 12->32 34 12->34 13->33 35 13->35 14->31 14->34 15->32 15->35 16->18 21 16->21 17->19 22 17->22 18->20 23 18->23 19->16 24 19->24 20->17 25 20->25 21->3 21->11 22->4 22->12 23->5 23->13 24->1 24->14 25->2 25->15 26->1 26->23 27->2 27->24 28->3 28->25 29->4 29->21 30->5 30->22 31->8 31->26 32->9 32->27 33->10 33->28 34->6 34->29 35->7 35->30
Degree $35$
Order $210$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times F_7$

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Copy content magma:G := TransitiveGroup(35, 10);
 

Group invariants

Abstract group:  $C_5\times F_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $210=2 \cdot 3 \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  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 $t$:  $10$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $5$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,28,25,2,29,21,3,30,22,4,26,23,5,27,24)(6,13,35,7,14,31,8,15,32,9,11,33,10,12,34)(16,18,20,17,19)$, $(1,16,21,11,31,26)(2,17,22,12,32,27)(3,18,23,13,33,28)(4,19,24,14,34,29)(5,20,25,15,35,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $C_6$
$10$:  $C_{10}$
$15$:  $C_{15}$
$30$:  $C_{30}$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Degree 7: $F_7$

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 $2^{15},1^{5}$ $7$ $2$ $15$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,31)(12,32)(13,33)(14,34)(15,35)(16,26)(17,27)(18,28)(19,29)(20,30)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,11,16)( 2,12,17)( 3,13,18)( 4,14,19)( 5,15,20)( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29)(10,35,30)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,16,11)( 2,17,12)( 3,18,13)( 4,19,14)( 5,20,15)( 6,26,31)( 7,27,32)( 8,28,33)( 9,29,34)(10,30,35)$
5A1 $5^{7}$ $1$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
5A-1 $5^{7}$ $1$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
5A2 $5^{7}$ $1$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
5A-2 $5^{7}$ $1$ $5$ $28$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
6A1 $6^{5},1^{5}$ $7$ $6$ $25$ $( 1,26,11, 6,16,31)( 2,27,12, 7,17,32)( 3,28,13, 8,18,33)( 4,29,14, 9,19,34)( 5,30,15,10,20,35)$
6A-1 $6^{5},1^{5}$ $7$ $6$ $25$ $( 1,31,16, 6,11,26)( 2,32,17, 7,12,27)( 3,33,18, 8,13,28)( 4,34,19, 9,14,29)( 5,35,20,10,15,30)$
7A $7^{5}$ $6$ $7$ $30$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
10A1 $10^{3},5$ $7$ $10$ $31$ $( 1, 9, 2,10, 3, 6, 4, 7, 5, 8)(11,34,12,35,13,31,14,32,15,33)(16,29,17,30,18,26,19,27,20,28)(21,24,22,25,23)$
10A-1 $10^{3},5$ $7$ $10$ $31$ $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,33,15,32,14,31,13,35,12,34)(16,28,20,27,19,26,18,30,17,29)(21,23,25,22,24)$
10A3 $10^{3},5$ $7$ $10$ $31$ $( 1,10, 4, 8, 2, 6, 5, 9, 3, 7)(11,35,14,33,12,31,15,34,13,32)(16,30,19,28,17,26,20,29,18,27)(21,25,24,23,22)$
10A-3 $10^{3},5$ $7$ $10$ $31$ $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,32,13,34,15,31,12,33,14,35)(16,27,18,29,20,26,17,28,19,30)(21,22,23,24,25)$
15A1 $15^{2},5$ $7$ $15$ $32$ $( 1,18,15, 2,19,11, 3,20,12, 4,16,13, 5,17,14)( 6,28,35, 7,29,31, 8,30,32, 9,26,33,10,27,34)(21,23,25,22,24)$
15A-1 $15^{2},5$ $7$ $15$ $32$ $( 1,14,17, 5,13,16, 4,12,20, 3,11,19, 2,15,18)( 6,34,27,10,33,26, 9,32,30, 8,31,29, 7,35,28)(21,24,22,25,23)$
15A2 $15^{2},5$ $7$ $15$ $32$ $( 1,15,19, 3,12,16, 5,14,18, 2,11,20, 4,13,17)( 6,35,29, 8,32,26,10,34,28, 7,31,30, 9,33,27)(21,25,24,23,22)$
15A-2 $15^{2},5$ $7$ $15$ $32$ $( 1,17,13, 4,20,11, 2,18,14, 5,16,12, 3,19,15)( 6,27,33, 9,30,31, 7,28,34,10,26,32, 8,29,35)(21,22,23,24,25)$
15A4 $15^{2},5$ $7$ $15$ $32$ $( 1,19,12, 5,18,11, 4,17,15, 3,16,14, 2,20,13)( 6,29,32,10,28,31, 9,27,35, 8,26,34, 7,30,33)(21,24,22,25,23)$
15A-4 $15^{2},5$ $7$ $15$ $32$ $( 1,13,20, 2,14,16, 3,15,17, 4,11,18, 5,12,19)( 6,33,30, 7,34,26, 8,35,27, 9,31,28,10,32,29)(21,23,25,22,24)$
15A7 $15^{2},5$ $7$ $15$ $32$ $( 1,20,14, 3,17,11, 5,19,13, 2,16,15, 4,18,12)( 6,30,34, 8,27,31,10,29,33, 7,26,35, 9,28,32)(21,25,24,23,22)$
15A-7 $15^{2},5$ $7$ $15$ $32$ $( 1,12,18, 4,15,16, 2,13,19, 5,11,17, 3,14,20)( 6,32,28, 9,35,26, 7,33,29,10,31,27, 8,34,30)(21,22,23,24,25)$
30A1 $30,5$ $7$ $30$ $33$ $( 1,32,18, 9,15,26, 2,33,19,10,11,27, 3,34,20, 6,12,28, 4,35,16, 7,13,29, 5,31,17, 8,14,30)(21,22,23,24,25)$
30A-1 $30,5$ $7$ $30$ $33$ $( 1,20,24,13,32,26, 5,19,23,12,31,30, 4,18,22,11,35,29, 3,17,21,15,34,28, 2,16,25,14,33,27)( 6,10, 9, 8, 7)$
30A7 $30,5$ $7$ $30$ $33$ $( 1,28,35,12,24,16, 3,30,32,14,21,18, 5,27,34,11,23,20, 2,29,31,13,25,17, 4,26,33,15,22,19)( 6, 8,10, 7, 9)$
30A-7 $30,5$ $7$ $30$ $33$ $( 1,24,17,30, 8,11, 4,22,20,28, 6,14, 2,25,18,26, 9,12, 5,23,16,29, 7,15, 3,21,19,27,10,13)(31,34,32,35,33)$
30A11 $30,5$ $7$ $30$ $33$ $( 1,32, 8,24,30,16, 2,33, 9,25,26,17, 3,34,10,21,27,18, 4,35, 6,22,28,19, 5,31, 7,23,29,20)(11,12,13,14,15)$
30A-11 $30,5$ $7$ $30$ $33$ $( 1,20,29,23, 7,31, 5,19,28,22, 6,35, 4,18,27,21,10,34, 3,17,26,25, 9,33, 2,16,30,24, 8,32)(11,15,14,13,12)$
30A13 $30,5$ $7$ $30$ $33$ $( 1,24,12,20,33, 6, 4,22,15,18,31, 9, 2,25,13,16,34, 7, 5,23,11,19,32,10, 3,21,14,17,35, 8)(26,29,27,30,28)$
30A-13 $30,5$ $7$ $30$ $33$ $( 1,28,15, 7,19,31, 3,30,12, 9,16,33, 5,27,14, 6,18,35, 2,29,11, 8,20,32, 4,26,13,10,17,34)(21,23,25,22,24)$
35A1 $35$ $6$ $35$ $34$ $( 1,32,28,24,20,11, 7, 3,34,30,21,17,13, 9, 5,31,27,23,19,15, 6, 2,33,29,25,16,12, 8, 4,35,26,22,18,14,10)$
35A-1 $35$ $6$ $35$ $34$ $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12,26,10,24, 3,17,31,15,29, 8,22)$
35A2 $35$ $6$ $35$ $34$ $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9,31,23,15, 2,29,16, 8,35,22,14)$
35A-2 $35$ $6$ $35$ $34$ $( 1,24, 7,30,13,31,19, 2,25, 8,26,14,32,20, 3,21, 9,27,15,33,16, 4,22,10,28,11,34,17, 5,23, 6,29,12,35,18)$

Malle's constant $a(G)$:     $1/15$

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Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed