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Properties

Label 35T40
35T40 1 17 1->17 33 1->33 2 16 2->16 32 2->32 3 20 3->20 31 3->31 4 19 4->19 35 4->35 5 18 5->18 34 5->34 6 6->2 23 6->23 7 7->1 22 7->22 8 8->5 21 8->21 9 9->4 25 9->25 10 10->3 24 10->24 11 11->3 12 11->12 12->2 13 13->1 15 13->15 14 14->5 15->4 28 16->28 16->32 27 17->27 17->31 26 18->26 18->35 30 19->30 19->34 29 20->29 20->33 21->27 22->26 23->30 24->29 25->28 31->7 31->13 32->6 32->12 33->10 33->11 34->9 34->15 35->8 35->14
Degree $35$
Order $25200$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $D_5\times A_7$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(35, 40);
 

Group invariants

Abstract group:  $D_5\times A_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$
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 $t$:  $40$
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:  $(1,33,11,3,31,13)(2,32,12)(4,35,14,5,34,15)(6,23)(7,22)(8,21)(9,25)(10,24)(16,28)(17,27)(18,26)(19,30)(20,29)$, $(1,17,31,7)(2,16,32,6)(3,20,33,10)(4,19,34,9)(5,18,35,8)(11,12)(13,15)(21,27)(22,26)(23,30)(24,29)(25,28)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$
$2520$:  $A_7$
$5040$:  $A_7\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $A_7$

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 $2^{14},1^{7}$ $5$ $2$ $14$ $( 1, 3)( 4, 5)( 6, 8)( 9,10)(11,13)(14,15)(16,18)(19,20)(21,23)(24,25)(26,28)(29,30)(31,33)(34,35)$
2B $2^{10},1^{15}$ $105$ $2$ $10$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)(16,26)(17,27)(18,28)(19,29)(20,30)$
2C $2^{16},1^{3}$ $525$ $2$ $16$ $( 1,34)( 2,33)( 3,32)( 4,31)( 5,35)( 6, 9)( 7, 8)(11,14)(12,13)(16,29)(17,28)(18,27)(19,26)(20,30)(21,24)(22,23)$
3A $3^{5},1^{20}$ $70$ $3$ $10$ $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)$
3B $3^{10},1^{5}$ $280$ $3$ $20$ $( 1,21,11)( 2,22,12)( 3,23,13)( 4,24,14)( 5,25,15)( 6,16,31)( 7,17,32)( 8,18,33)( 9,19,34)(10,20,35)$
4A $4^{5},2^{5},1^{5}$ $630$ $4$ $20$ $( 1, 6,26,21)( 2, 7,27,22)( 3, 8,28,23)( 4, 9,29,24)( 5,10,30,25)(11,31)(12,32)(13,33)(14,34)(15,35)$
4B $4^{5},2^{7},1$ $3150$ $4$ $22$ $( 1,35)( 2,34)( 3,33)( 4,32)( 5,31)( 6,15,26,20)( 7,14,27,19)( 8,13,28,18)( 9,12,29,17)(10,11,30,16)(21,25)(22,24)$
5A1 $5^{7}$ $2$ $5$ $28$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
5A2 $5^{7}$ $2$ $5$ $28$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
5B $5^{5},1^{10}$ $504$ $5$ $20$ $(11,21,16,31,26)(12,22,17,32,27)(13,23,18,33,28)(14,24,19,34,29)(15,25,20,35,30)$
5C1 $5^{7}$ $1008$ $5$ $28$ $( 1, 9,27,35,13)( 2,10,28,31,14)( 3, 6,29,32,15)( 4, 7,30,33,11)( 5, 8,26,34,12)(16,19,17,20,18)(21,24,22,25,23)$
5C2 $5^{7}$ $1008$ $5$ $28$ $( 1,27,13, 9,35)( 2,28,14,10,31)( 3,29,15, 6,32)( 4,30,11, 7,33)( 5,26,12, 8,34)(16,17,18,19,20)(21,22,23,24,25)$
6A $3^{5},2^{10}$ $210$ $6$ $20$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,26)(17,27)(18,28)(19,29)(20,30)$
6B $6^{2},3,2^{8},1^{4}$ $350$ $6$ $20$ $( 1,24,16, 4,21,19)( 2,23,17, 3,22,18)( 5,25,20)( 6, 9)( 7, 8)(11,14)(12,13)(26,29)(27,28)(31,34)(32,33)$
6C $6^{2},3,2^{10}$ $1050$ $6$ $22$ $( 1,34)( 2,33)( 3,32)( 4,31)( 5,35)( 6,14,21, 9,11,24)( 7,13,22, 8,12,23)(10,15,25)(16,29)(17,28)(18,27)(19,26)(20,30)$
6D $6^{4},3^{2},2^{2},1$ $1400$ $6$ $26$ $( 1,13, 6, 3,11, 8)( 2,12, 7)( 4,15, 9, 5,14,10)(16,23,26,18,21,28)(17,22,27)(19,25,29,20,24,30)(31,33)(34,35)$
7A1 $7^{5}$ $360$ $7$ $30$ $( 1,11,31,26, 6,16,21)( 2,12,32,27, 7,17,22)( 3,13,33,28, 8,18,23)( 4,14,34,29, 9,19,24)( 5,15,35,30,10,20,25)$
7A-1 $7^{5}$ $360$ $7$ $30$ $( 1,21,16, 6,26,31,11)( 2,22,17, 7,27,32,12)( 3,23,18, 8,28,33,13)( 4,24,19, 9,29,34,14)( 5,25,20,10,30,35,15)$
10A1 $10^{2},5^{3}$ $210$ $10$ $30$ $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,30,19,28,17,26,20,29,18,27)(21,25,24,23,22)$
10A3 $10^{2},5^{3}$ $210$ $10$ $30$ $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,28,20,27,19,26,18,30,17,29)(21,23,25,22,24)$
10B $10^{2},5,2^{4},1^{2}$ $2520$ $10$ $26$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,35,21,30,16,15,31,25,26,20)(12,34,22,29,17,14,32,24,27,19)(13,33,23,28,18)$
14A1 $14^{2},7$ $1800$ $14$ $32$ $( 1, 8,11,18,31,23,26, 3, 6,13,16,33,21,28)( 2, 7,12,17,32,22,27)( 4,10,14,20,34,25,29, 5, 9,15,19,35,24,30)$
14A-1 $14^{2},7$ $1800$ $14$ $32$ $( 1,28,21,33,16,13, 6, 3,26,23,31,18,11, 8)( 2,27,22,32,17,12, 7)( 4,30,24,35,19,15, 9, 5,29,25,34,20,14,10)$
15A1 $15,5^{4}$ $140$ $15$ $30$ $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,17,18,19,20)(26,27,28,29,30)(31,32,33,34,35)$
15A2 $15,5^{4}$ $140$ $15$ $30$ $( 1, 3, 5, 2, 4)( 6,23,15, 7,24,11, 8,25,12, 9,21,13,10,22,14)(16,18,20,17,19)(26,28,30,27,29)(31,33,35,32,34)$
15B1 $15^{2},5$ $560$ $15$ $32$ $( 1,15,24, 3,12,21, 5,14,23, 2,11,25, 4,13,22)( 6,35,19, 8,32,16,10,34,18, 7,31,20, 9,33,17)(26,30,29,28,27)$
15B2 $15^{2},5$ $560$ $15$ $32$ $( 1,24,12, 5,23,11, 4,22,15, 3,21,14, 2,25,13)( 6,19,32,10,18,31, 9,17,35, 8,16,34, 7,20,33)(26,29,27,30,28)$
20A1 $20,10,5$ $1260$ $20$ $32$ $( 1, 7,28,24, 5, 6,27,23, 4,10,26,22, 3, 9,30,21, 2, 8,29,25)(11,32,13,34,15,31,12,33,14,35)(16,17,18,19,20)$
20A3 $20,10,5$ $1260$ $20$ $32$ $( 1,24,27,10, 3,21,29, 7, 5,23,26, 9, 2,25,28, 6, 4,22,30, 8)(11,34,12,35,13,31,14,32,15,33)(16,19,17,20,18)$
30A1 $15,10^{2}$ $420$ $30$ $32$ $( 1,34, 2,35, 3,31, 4,32, 5,33)( 6,24,12,10,23,11, 9,22,15, 8,21,14, 7,25,13)(16,29,17,30,18,26,19,27,20,28)$
30A7 $15,10^{2}$ $420$ $30$ $32$ $( 1,32, 3,34, 5,31, 2,33, 4,35)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,18,29,20,26,17,28,19,30)$
35A1 $35$ $720$ $35$ $34$ $( 1,35,19, 8,12,26,25, 4,33,17, 6,15,29,23, 2,31,20, 9,13,27,21, 5,34,18, 7,11,30,24, 3,32,16,10,14,28,22)$
35A-1 $35$ $720$ $35$ $34$ $( 1,22,28,14,10,16,32, 3,24,30,11, 7,18,34, 5,21,27,13, 9,20,31, 2,23,29,15, 6,17,33, 4,25,26,12, 8,19,35)$
35A2 $35$ $720$ $35$ $34$ $( 1,19,12,25,33, 6,29, 2,20,13,21,34, 7,30, 3,16,14,22,35, 8,26, 4,17,15,23,31, 9,27, 5,18,11,24,32,10,28)$
35A-2 $35$ $720$ $35$ $34$ $( 1,28,10,32,24,11,18, 5,27, 9,31,23,15,17, 4,26, 8,35,22,14,16, 3,30, 7,34,21,13,20, 2,29, 6,33,25,12,19)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed