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Properties

Label 35T22
35T22 1 15 1->15 28 1->28 2 13 2->13 26 2->26 3 11 3->11 30 3->30 4 14 4->14 27 4->27 5 12 5->12 29 5->29 6 6->12 20 6->20 7 7->13 19 7->19 8 8->15 17 8->17 9 9->11 16 9->16 10 10->14 18 10->18 25 11->25 33 11->33 22 12->22 31 12->31 21 13->21 35 13->35 23 14->23 34 14->34 24 15->24 32 15->32 16->19 16->27 17->29 18->16 18->26 19->18 19->28 20->30 21->5 21->35 22->3 22->32 23->4 23->31 24->1 24->34 25->2 25->33 26->5 26->25 27->3 27->23 28->2 28->24 29->4 29->21 30->1 30->22 31->6 31->9 32->8 32->10 33->8 33->9 34->7 34->10 35->6 35->7
Degree $35$
Order $1260$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_7:\GL(2,4)$

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Copy content magma:G := TransitiveGroup(35, 22);
 

Group invariants

Abstract group:  $C_7:\GL(2,4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $1260=2^{2} \cdot 3^{2} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  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 $t$:  $22$
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:  $(1,15,24,34,7,19,28,2,13,21,35,6,20,30)(3,11,25,33,8,17,29,4,14,23,31,9,16,27)(5,12,22,32,10,18,26)$, $(1,28,24)(2,26,25)(3,30,22)(4,27,23)(5,29,21)(6,12,31)(7,13,35)(8,15,32)(9,11,33)(10,14,34)(16,19,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$21$:  $C_7:C_3$
$60$:  $A_5$
$180$:  $\GL(2,4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $A_5$

Degree 7: $C_7:C_3$

Low degree siblings

42T197

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 $2^{14},1^{7}$ $15$ $2$ $14$ $( 1, 4)( 2, 5)( 6,10)( 7, 9)(11,13)(12,15)(17,20)(18,19)(21,22)(23,24)(26,30)(27,28)(32,34)(33,35)$
3A1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,24,35)( 2,21,34)( 3,25,31)( 4,23,33)( 5,22,32)(11,27,17)(12,26,18)(13,28,20)(14,29,16)(15,30,19)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 1,35,24)( 2,34,21)( 3,31,25)( 4,33,23)( 5,32,22)(11,17,27)(12,18,26)(13,20,28)(14,16,29)(15,19,30)$
3B $3^{7},1^{14}$ $20$ $3$ $14$ $( 1, 4, 3)( 7, 9, 8)(11,14,13)(16,20,17)(23,25,24)(27,29,28)(31,35,33)$
3C1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 3, 4, 5)( 6,21,15)( 7,24,13)( 8,23,12)( 9,22,14)(10,25,11)(16,27,32)(17,26,31)(18,29,33)(19,30,34)(20,28,35)$
3C-1 $3^{11},1^{2}$ $140$ $3$ $22$ $( 3, 5, 4)( 6,15,21)( 7,13,24)( 8,12,23)( 9,14,22)(10,11,25)(16,32,27)(17,31,26)(18,33,29)(19,34,30)(20,35,28)$
5A1 $5^{7}$ $12$ $5$ $28$ $( 1, 5, 3, 4, 2)( 6, 7,10, 8, 9)(11,15,13,12,14)(16,17,19,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,33,34,35,32)$
5A2 $5^{7}$ $12$ $5$ $28$ $( 1, 3, 2, 5, 4)( 6,10, 9, 7, 8)(11,13,14,15,12)(16,19,18,17,20)(21,22,23,24,25)(26,27,28,29,30)(31,34,32,33,35)$
6A1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1,33,24, 4,35,23)( 2,32,21, 5,34,22)( 3,31,25)( 6,10)( 7, 9)(11,20,27,13,17,28)(12,19,26,15,18,30)(14,16,29)$
6A-1 $6^{4},3^{2},2^{2},1$ $105$ $6$ $26$ $( 1,23,35, 4,24,33)( 2,22,34, 5,21,32)( 3,25,31)( 6,10)( 7, 9)(11,28,17,13,27,20)(12,30,18,15,26,19)(14,29,16)$
7A1 $7^{5}$ $3$ $7$ $30$ $( 1, 7,13,20,24,28,35)( 2, 6,15,19,21,30,34)( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)( 5,10,12,18,22,26,32)$
7A-1 $7^{5}$ $3$ $7$ $30$ $( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8)( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
14A1 $14^{2},7$ $45$ $14$ $32$ $( 1,21, 7,30,13,34,20, 2,24, 6,28,15,35,19)( 3,23, 8,27,14,33,16, 4,25, 9,29,11,31,17)( 5,22,10,26,12,32,18)$
14A-1 $14^{2},7$ $45$ $14$ $32$ $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,17,31,11,29, 9,25, 4,16,33,14,27, 8,23)( 5,18,32,12,26,10,22)$
15A1 $15^{2},5$ $84$ $15$ $32$ $( 1,29, 9, 5,30, 7, 3,27,10, 2,28, 8, 4,26, 6)(11,12,15,13,14)(16,23,32,19,24,31,17,22,34,20,25,33,18,21,35)$
15A-1 $15^{2},5$ $84$ $15$ $32$ $( 1, 6,26, 4, 8,28, 2,10,27, 3, 7,30, 5, 9,29)(11,14,13,15,12)(16,35,21,18,33,25,20,34,22,17,31,24,19,32,23)$
15A2 $15^{2},5$ $84$ $15$ $32$ $( 1, 9,30, 3,10,28, 4, 6,29, 5, 7,27, 2, 8,26)(11,15,14,12,13)(16,32,24,17,34,25,18,35,23,19,31,22,20,33,21)$
15A-2 $15^{2},5$ $84$ $15$ $32$ $( 1,26, 8, 2,27, 7, 5,29, 6, 4,28,10, 3,30, 9)(11,13,12,14,15)(16,21,33,20,22,31,19,23,35,18,25,34,17,24,32)$
21A1 $21,7^{2}$ $60$ $21$ $32$ $( 1, 9,14,20,23,29,35, 4, 8,13,17,25,28,33, 3, 7,11,16,24,27,31)( 2, 6,15,19,21,30,34)( 5,10,12,18,22,26,32)$
21A-1 $21,7^{2}$ $60$ $21$ $32$ $( 1,31,27,24,16,11, 7, 3,33,28,25,17,13, 8, 4,35,29,23,20,14, 9)( 2,34,30,21,19,15, 6)( 5,32,26,22,18,12,10)$
35A1 $35$ $36$ $35$ $34$ $( 1, 9,12,19,25,28,33, 5, 6,14,20,23,26,34, 3, 7,11,18,21,29,35, 4,10,15,16,24,27,32, 2, 8,13,17,22,30,31)$
35A-1 $35$ $36$ $35$ $34$ $( 1,31,30,22,17,13, 8, 2,32,27,24,16,15,10, 4,35,29,21,18,11, 7, 3,34,26,23,20,14, 6, 5,33,28,25,19,12, 9)$
35A2 $35$ $36$ $35$ $34$ $( 1,12,25,33, 6,20,26, 3,11,21,35,10,16,27, 2,13,22,31, 9,19,28, 5,14,23,34, 7,18,29, 4,15,24,32, 8,17,30)$
35A-2 $35$ $36$ $35$ $34$ $( 1,30,17, 8,32,24,15, 4,29,18, 7,34,23,14, 5,28,19, 9,31,22,13, 2,27,16,10,35,21,11, 3,26,20, 6,33,25,12)$

Malle's constant $a(G)$:     $1/14$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B 3C1 3C-1 5A1 5A2 6A1 6A-1 7A1 7A-1 14A1 14A-1 15A1 15A-1 15A2 15A-2 21A1 21A-1 35A1 35A-1 35A2 35A-2
Size 1 15 7 7 20 140 140 12 12 105 105 3 3 45 45 84 84 84 84 60 60 36 36 36 36
2 P 1A 1A 3A-1 3A1 3B 3C-1 3C1 5A2 5A1 3A1 3A-1 7A1 7A-1 7A1 7A-1 15A2 15A-2 15A1 15A-1 21A1 21A-1 35A2 35A-2 35A1 35A-1
3 P 1A 2A 1A 1A 1A 1A 1A 5A2 5A1 2A 2A 7A-1 7A1 14A-1 14A1 5A2 5A2 5A1 5A1 7A-1 7A1 35A-2 35A2 35A-1 35A1
5 P 1A 2A 3A-1 3A1 3B 3C-1 3C1 1A 1A 6A-1 6A1 7A-1 7A1 14A-1 14A1 3A1 3A-1 3A-1 3A1 21A-1 21A1 7A-1 7A1 7A-1 7A1
7 P 1A 2A 3A1 3A-1 3B 3C1 3C-1 5A2 5A1 6A1 6A-1 1A 1A 2A 2A 15A-2 15A2 15A-1 15A1 3B 3B 5A1 5A1 5A2 5A2
Type
1260.61.1a 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
1260.61.1b1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3 ζ3−1 ζ3−1 1 1 1 1 1 1
1260.61.1b2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3−1 ζ3 ζ3 1 1 1 1 1 1
1260.61.3a1 C 3 3 0 0 3 0 0 3 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
1260.61.3a2 C 3 3 0 0 3 0 0 3 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
1260.61.3b1 R 3 −1 3 3 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 −1 3 3 −1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
1260.61.3b2 R 3 −1 3 3 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 −1 3 3 −1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
1260.61.3c1 C 3 −1 3ζ15−5 3ζ155 0 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ155 −ζ15−5 3 3 −1 −1 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −1+ζ15+ζ154−ζ155 −ζ15−ζ154 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−6−ζ156
1260.61.3c2 C 3 −1 3ζ155 3ζ15−5 0 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−5 −ζ155 3 3 −1 −1 −1+ζ15+ζ154−ζ155 −ζ15−ζ154 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ152−ζ153+ζ154+ζ157 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−6−ζ156
1260.61.3c3 C 3 −1 3ζ15−5 3ζ155 0 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ155 −ζ15−5 3 3 −1 −1 −1+ζ15+ζ152−ζ153+ζ154+ζ157 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −ζ15−ζ154 −1+ζ15+ζ154−ζ155 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−3−ζ153
1260.61.3c4 C 3 −1 3ζ155 3ζ15−5 0 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−5 −ζ155 3 3 −1 −1 −ζ15−ζ154 −1+ζ15+ζ154−ζ155 −1+ζ15+ζ152−ζ153+ζ154+ζ157 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−3−ζ153
1260.61.4a R 4 0 4 4 1 1 1 −1 −1 0 0 4 4 0 0 −1 −1 −1 −1 1 1 −1 −1 −1 −1
1260.61.4b1 C 4 0 4ζ3−1 4ζ3 1 ζ3−1 ζ3 −1 −1 0 0 4 4 0 0 −ζ3 −ζ3 −ζ3−1 −ζ3−1 1 1 −1 −1 −1 −1
1260.61.4b2 C 4 0 4ζ3 4ζ3−1 1 ζ3 ζ3−1 −1 −1 0 0 4 4 0 0 −ζ3−1 −ζ3−1 −ζ3 −ζ3 1 1 −1 −1 −1 −1
1260.61.5a R 5 1 5 5 −1 −1 −1 0 0 1 1 5 5 1 1 0 0 0 0 −1 −1 0 0 0 0
1260.61.5b1 C 5 1 5ζ3−1 5ζ3 −1 −ζ3−1 −ζ3 0 0 ζ3 ζ3−1 5 5 1 1 0 0 0 0 −1 −1 0 0 0 0
1260.61.5b2 C 5 1 5ζ3 5ζ3−1 −1 −ζ3 −ζ3−1 0 0 ζ3−1 ζ3 5 5 1 1 0 0 0 0 −1 −1 0 0 0 0
1260.61.9a1 C 9 −3 0 0 0 0 0 −3ζ35−7−3ζ357 −3ζ35−14−3ζ3514 0 0 −3ζ35−15−3−3ζ355−3ζ3510 3ζ35−15+3ζ355+3ζ3510 ζ35−15+1+ζ355+ζ3510 −ζ35−15−ζ355−ζ3510 0 0 0 0 0 0 −ζ35−14+ζ35−13−ζ35−10−ζ35−5−1−ζ354+ζ358−ζ359−ζ3511−ζ3514−ζ3516 ζ35−13−ζ354+ζ358−ζ359−ζ3511+ζ3515−ζ3516 −ζ35−13+ζ35−10+ζ35−5+ζ354−ζ358+ζ359+ζ3511+ζ3516 ζ35−14−ζ35−13+ζ354−ζ358+ζ359+ζ3511+ζ3514−ζ3515+ζ3516
1260.61.9a2 C 9 −3 0 0 0 0 0 −3ζ35−7−3ζ357 −3ζ35−14−3ζ3514 0 0 3ζ35−15+3ζ355+3ζ3510 −3ζ35−15−3−3ζ355−3ζ3510 −ζ35−15−ζ355−ζ3510 ζ35−15+1+ζ355+ζ3510 0 0 0 0 0 0 −ζ35−13+ζ35−10+ζ35−5+ζ354−ζ358+ζ359+ζ3511+ζ3516 ζ35−14−ζ35−13+ζ354−ζ358+ζ359+ζ3511+ζ3514−ζ3515+ζ3516 −ζ35−14+ζ35−13−ζ35−10−ζ35−5−1−ζ354+ζ358−ζ359−ζ3511−ζ3514−ζ3516 ζ35−13−ζ354+ζ358−ζ359−ζ3511+ζ3515−ζ3516
1260.61.9a3 C 9 −3 0 0 0 0 0 −3ζ35−14−3ζ3514 −3ζ35−7−3ζ357 0 0 −3ζ35−15−3−3ζ355−3ζ3510 3ζ35−15+3ζ355+3ζ3510 ζ35−15+1+ζ355+ζ3510 −ζ35−15−ζ355−ζ3510 0 0 0 0 0 0 ζ35−14−ζ35−13+ζ354−ζ358+ζ359+ζ3511+ζ3514−ζ3515+ζ3516 −ζ35−13+ζ35−10+ζ35−5+ζ354−ζ358+ζ359+ζ3511+ζ3516 ζ35−13−ζ354+ζ358−ζ359−ζ3511+ζ3515−ζ3516 −ζ35−14+ζ35−13−ζ35−10−ζ35−5−1−ζ354+ζ358−ζ359−ζ3511−ζ3514−ζ3516
1260.61.9a4 C 9 −3 0 0 0 0 0 −3ζ35−14−3ζ3514 −3ζ35−7−3ζ357 0 0 3ζ35−15+3ζ355+3ζ3510 −3ζ35−15−3−3ζ355−3ζ3510 −ζ35−15−ζ355−ζ3510 ζ35−15+1+ζ355+ζ3510 0 0 0 0 0 0 ζ35−13−ζ354+ζ358−ζ359−ζ3511+ζ3515−ζ3516 −ζ35−14+ζ35−13−ζ35−10−ζ35−5−1−ζ354+ζ358−ζ359−ζ3511−ζ3514−ζ3516 ζ35−14−ζ35−13+ζ354−ζ358+ζ359+ζ3511+ζ3514−ζ3515+ζ3516 −ζ35−13+ζ35−10+ζ35−5+ζ354−ζ358+ζ359+ζ3511+ζ3516
1260.61.12a1 C 12 0 0 0 3 0 0 −3 −3 0 0 −4ζ7−3−4−4ζ7−4ζ72 4ζ7−3+4ζ7+4ζ72 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
1260.61.12a2 C 12 0 0 0 3 0 0 −3 −3 0 0 4ζ7−3+4ζ7+4ζ72 −4ζ7−3−4−4ζ7−4ζ72 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
1260.61.15a1 C 15 3 0 0 −3 0 0 0 0 0 0 −5ζ7−3−5−5ζ7−5ζ72 5ζ7−3+5ζ7+5ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 0 0 0 0
1260.61.15a2 C 15 3 0 0 −3 0 0 0 0 0 0 5ζ7−3+5ζ7+5ζ72 −5ζ7−3−5−5ζ7−5ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed