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Properties

Label 35T9
35T9 1 12 1->12 35 1->35 2 11 2->11 31 2->31 3 15 3->15 32 3->32 4 14 4->14 33 4->33 5 13 5->13 34 5->34 6 17 6->17 20 6->20 7 16 7->16 7->16 8 8->17 8->20 9 18 9->18 19 9->19 10 10->18 10->19 11->5 22 11->22 12->1 21 12->21 13->2 25 13->25 14->3 24 14->24 15->4 23 15->23 16->25 27 16->27 17->21 26 17->26 18->22 30 18->30 19->23 29 19->29 20->24 28 20->28 21->10 21->32 22->6 22->31 23->7 23->35 24->8 24->34 25->9 25->33 26->2 26->30 27->1 27->26 28->5 28->27 29->4 29->28 30->3 30->29 31->7 31->15 32->6 32->11 33->10 33->12 34->9 34->13 35->8 35->14
Degree $35$
Order $210$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{35}:C_6$

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Copy content magma:G := TransitiveGroup(35, 9);
 

Group invariants

Abstract group:  $C_{35}:C_6$
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 $t$:  $9$
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:  $(1,35,14,3,32,11,5,34,13,2,31,15,4,33,12)(6,20,24,8,17,21,10,19,23,7,16,25,9,18,22)(26,30,29,28,27)$, $(1,12,21,32,6,17,26,2,11,22,31,7,16,27)(3,15,23,35,8,20,28,5,13,25,33,10,18,30)(4,14,24,34,9,19,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$
$21$:  $C_7:C_3$
$30$:  $D_5\times C_3$
$42$:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $C_7:C_3$

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

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 5A1 5A2 6A1 6A-1 7A1 7A-1 14A1 14A-1 15A1 15A-1 15A2 15A-2 35A1 35A-1 35A2 35A-2
Size 1 5 7 7 2 2 35 35 3 3 15 15 14 14 14 14 6 6 6 6
2 P 1A 1A 3A-1 3A1 5A2 5A1 3A-1 3A1 7A1 7A-1 7A-1 7A1 15A2 15A-2 15A1 15A-1 35A2 35A-2 35A1 35A-1
3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 7A-1 7A1 14A-1 14A1 5A1 5A1 5A2 5A2 35A-2 35A2 35A-1 35A1
5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 7A-1 7A1 14A-1 14A1 3A1 3A-1 3A-1 3A1 7A1 7A-1 7A1 7A-1
7 P 1A 2A 3A1 3A-1 5A2 5A1 6A1 6A-1 1A 1A 2A 2A 15A-2 15A2 15A-1 15A1 5A1 5A1 5A2 5A2
Type
210.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
210.2.1b R 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1
210.2.1c1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
210.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.2.1d1 C 1 −1 ζ3−1 ζ3 1 1 −ζ3−1 −ζ3 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
210.2.1d2 C 1 −1 ζ3 ζ3−1 1 1 −ζ3 −ζ3−1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.2.2a1 R 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 2 2 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
210.2.2a2 R 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 2 2 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
210.2.2b1 C 2 0 2ζ15−5 2ζ155 ζ15−6+ζ156 ζ15−3+ζ153 0 0 2 2 0 0 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153
210.2.2b2 C 2 0 2ζ155 2ζ15−5 ζ15−6+ζ156 ζ15−3+ζ153 0 0 2 2 0 0 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153
210.2.2b3 C 2 0 2ζ15−5 2ζ155 ζ15−3+ζ153 ζ15−6+ζ156 0 0 2 2 0 0 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156
210.2.2b4 C 2 0 2ζ155 2ζ15−5 ζ15−3+ζ153 ζ15−6+ζ156 0 0 2 2 0 0 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156
210.2.3a1 C 3 3 0 0 3 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
210.2.3a2 C 3 3 0 0 3 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
210.2.3b1 C 3 −3 0 0 3 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
210.2.3b2 C 3 −3 0 0 3 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
210.2.6a1 C 6 0 0 0 3ζ35−14+3ζ3514 3ζ35−7+3ζ357 0 0 −2ζ35−15−2−2ζ355−2ζ3510 2ζ35−15+2ζ355+2ζ3510 0 0 0 0 0 0 −ζ35−13+ζ354−ζ358+ζ359+ζ3511−ζ3515+ζ3516 −ζ35−14+ζ35−13−ζ354+ζ358−ζ359−ζ3511−ζ3514+ζ3515−ζ3516 ζ35−13−ζ35−10−ζ35−5−ζ354+ζ358−ζ359−ζ3511−ζ3516 ζ35−14−ζ35−13+ζ35−10+ζ35−5+1+ζ354−ζ358+ζ359+ζ3511+ζ3514+ζ3516
210.2.6a2 C 6 0 0 0 3ζ35−14+3ζ3514 3ζ35−7+3ζ357 0 0 2ζ35−15+2ζ355+2ζ3510 −2ζ35−15−2−2ζ355−2ζ3510 0 0 0 0 0 0 −ζ35−14+ζ35−13−ζ354+ζ358−ζ359−ζ3511−ζ3514+ζ3515−ζ3516 −ζ35−13+ζ354−ζ358+ζ359+ζ3511−ζ3515+ζ3516 ζ35−14−ζ35−13+ζ35−10+ζ35−5+1+ζ354−ζ358+ζ359+ζ3511+ζ3514+ζ3516 ζ35−13−ζ35−10−ζ35−5−ζ354+ζ358−ζ359−ζ3511−ζ3516
210.2.6a3 C 6 0 0 0 3ζ35−7+3ζ357 3ζ35−14+3ζ3514 0 0 −2ζ35−15−2−2ζ355−2ζ3510 2ζ35−15+2ζ355+2ζ3510 0 0 0 0 0 0 ζ35−13−ζ35−10−ζ35−5−ζ354+ζ358−ζ359−ζ3511−ζ3516 ζ35−14−ζ35−13+ζ35−10+ζ35−5+1+ζ354−ζ358+ζ359+ζ3511+ζ3514+ζ3516 −ζ35−13+ζ354−ζ358+ζ359+ζ3511−ζ3515+ζ3516 −ζ35−14+ζ35−13−ζ354+ζ358−ζ359−ζ3511−ζ3514+ζ3515−ζ3516
210.2.6a4 C 6 0 0 0 3ζ35−7+3ζ357 3ζ35−14+3ζ3514 0 0 2ζ35−15+2ζ355+2ζ3510 −2ζ35−15−2−2ζ355−2ζ3510 0 0 0 0 0 0 ζ35−14−ζ35−13+ζ35−10+ζ35−5+1+ζ354−ζ358+ζ359+ζ3511+ζ3514+ζ3516 ζ35−13−ζ35−10−ζ35−5−ζ354+ζ358−ζ359−ζ3511−ζ3516 −ζ35−14+ζ35−13−ζ354+ζ358−ζ359−ζ3511−ζ3514+ζ3515−ζ3516 −ζ35−13+ζ354−ζ358+ζ359+ζ3511−ζ3515+ζ3516

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed