↑

Properties

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

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(35, 9);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{35}:C_6$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(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

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  210.2
magma: IdentifyGroup(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 3A1 3A-1 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);