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Properties

Label 35T9
Degree $35$
Order $210$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{35}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(35, 9);
 

Group invariants

Abstract group:  $C_{35}:C_6$
magma: IdentifyGroup(G);
 
Order:  $210=2 \cdot 3 \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,35,14,3,32,11,5,34,13,2,31,15,4,33,12)(6,20,24,8,17,21,10,19,23,7,16,25,9,18,22)(26,30,29,28,27)$, $(1,12,21,32,6,17,26,2,11,22,31,7,16,27)(3,15,23,35,8,20,28,5,13,25,33,10,18,30)(4,14,24,34,9,19,29)$
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 $3^{10},1^{5}$ $7$ $3$ $20$ $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27)(18,33,28)(19,34,29)(20,35,30)$
3A-1 $3^{10},1^{5}$ $7$ $3$ $20$ $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32)(18,28,33)(19,29,34)(20,30,35)$
5A1 $5^{7}$ $2$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
5A2 $5^{7}$ $2$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
6A1 $6^{4},3^{2},2^{2},1$ $35$ $6$ $26$ $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26)(17,35,27,20,32,30)(18,34,28,19,33,29)$
6A-1 $6^{4},3^{2},2^{2},1$ $35$ $6$ $26$ $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31)(17,30,32,20,27,35)(18,29,33,19,28,34)$
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 $7^{5}$ $3$ $7$ $30$ $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13)( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$
14A1 $14^{2},7$ $15$ $14$ $32$ $( 1,26,16, 6,31,21,11)( 2,30,17,10,32,25,12, 5,27,20, 7,35,22,15)( 3,29,18, 9,33,24,13, 4,28,19, 8,34,23,14)$
14A-1 $14^{2},7$ $15$ $14$ $32$ $( 1,21, 6,26,11,31,16)( 2,25, 7,30,12,35,17, 5,22,10,27,15,32,20)( 3,24, 8,29,13,34,18, 4,23, 9,28,14,33,19)$
15A1 $15^{2},5$ $14$ $15$ $32$ $( 1,18,15, 2,19,11, 3,20,12, 4,16,13, 5,17,14)( 6,28,35, 7,29,31, 8,30,32, 9,26,33,10,27,34)(21,23,25,22,24)$
15A-1 $15^{2},5$ $14$ $15$ $32$ $( 1,33,15, 2,34,11, 3,35,12, 4,31,13, 5,32,14)( 6,18,25, 7,19,21, 8,20,22, 9,16,23,10,17,24)(26,28,30,27,29)$
15A2 $15^{2},5$ $14$ $15$ $32$ $( 1,30,24, 3,27,21, 5,29,23, 2,26,25, 4,28,22)( 6,15,34, 8,12,31,10,14,33, 7,11,35, 9,13,32)(16,20,19,18,17)$
15A-2 $15^{2},5$ $14$ $15$ $32$ $( 1,35,24, 3,32,21, 5,34,23, 2,31,25, 4,33,22)( 6,10, 9, 8, 7)(11,20,29,13,17,26,15,19,28,12,16,30,14,18,27)$
35A1 $35$ $6$ $35$ $34$ $( 1,25, 9,28,12,31,20, 4,23, 7,26,15,34,18, 2,21,10,29,13,32,16, 5,24, 8,27,11,35,19, 3,22, 6,30,14,33,17)$
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 $35$ $6$ $35$ $34$ $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$
35A-2 $35$ $6$ $35$ $34$ $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9,31,23,15, 2,29,16, 8,35,22,14)$

Malle's constant $a(G)$:     $1/14$

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 15A-2 15A2 15A-1 15A1 35A-1 35A1 35A-2 35A2
3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 7A-1 7A1 14A-1 14A1 5A1 5A1 5A2 5A2 35A1 35A-1 35A2 35A-2
5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 7A-1 7A1 14A-1 14A1 3A-1 3A1 3A1 3A-1 7A-1 7A1 7A-1 7A1
7 P 1A 2A 3A1 3A-1 5A2 5A1 6A1 6A-1 1A 1A 2A 2A 15A2 15A-2 15A1 15A-1 5A2 5A2 5A1 5A1
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 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.2.1c2 C 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
210.2.1d1 C 1 −1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
210.2.1d2 C 1 −1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 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 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
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 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
210.2.2b1 C 2 0 2ζ15−5 2ζ155 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 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156
210.2.2b2 C 2 0 2ζ155 2ζ15−5 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 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156
210.2.2b3 C 2 0 2ζ15−5 2ζ155 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 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153
210.2.2b4 C 2 0 2ζ155 2ζ15−5 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 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153
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+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−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 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+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+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−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 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+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 ζ35−14−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
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 ζ35−13−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
210.2.6a3 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 −ζ35−14+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
210.2.6a4 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 −ζ35−13+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

magma: CharacterTable(G);
 

Regular extensions

Data not computed