↑

Properties

Label 35T5
Degree $35$
Order $105$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:C_{15}$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(35, 5);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:C_{15}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11,16)(2,12,17)(3,13,18)(4,14,19)(5,15,20)(6,31,26)(7,32,27)(8,33,28)(9,34,29)(10,35,30), (1,29,7,5,28,6,4,27,10,3,26,9,2,30,8)(11,14,12,15,13)(16,24,32,20,23,31,19,22,35,18,21,34,17,25,33)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $105=3 \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  105.1
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 5A1 5A-1 5A2 5A-2 7A1 7A-1 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 35A1 35A-1 35A2 35A-2 35A4 35A-4 35A8 35A-8
Size 1 7 7 1 1 1 1 3 3 7 7 7 7 7 7 7 7 3 3 3 3 3 3 3 3
3 P 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 7A1 7A-1 15A2 15A4 15A1 15A-1 15A-7 15A-4 15A7 15A-2 35A-4 35A-8 35A4 35A-1 35A8 35A1 35A-2 35A2
5 P 1A 1A 1A 5A-2 5A2 5A1 5A-1 7A-1 7A1 5A1 5A2 5A-2 5A2 5A-1 5A-2 5A1 5A-1 35A4 35A8 35A-4 35A1 35A-8 35A-1 35A2 35A-2
7 P 1A 3A-1 3A1 1A 1A 1A 1A 7A-1 7A1 3A1 3A-1 3A-1 3A1 3A1 3A1 3A-1 3A-1 7A-1 7A-1 7A1 7A-1 7A1 7A1 7A-1 7A1
Type
105.1.1a 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
105.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1 1
105.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1 1
105.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5
105.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1
105.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2
105.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52
105.1.1d1 C 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 1 1 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154 ζ15−6 ζ156 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−3 ζ153
105.1.1d2 C 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 1 1 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−6 ζ156 ζ153 ζ15−3
105.1.1d3 C 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 1 1 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−6 ζ156 ζ153 ζ15−3
105.1.1d4 C 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 1 1 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1 ζ15−6 ζ156 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−3 ζ153
105.1.1d5 C 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 1 1 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157 ζ15−3 ζ153 ζ15−6 ζ156 ζ153 ζ15−3 ζ156 ζ15−6
105.1.1d6 C 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 1 1 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−6 ζ156
105.1.1d7 C 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 1 1 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−6 ζ156
105.1.1d8 C 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 1 1 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152 ζ15−3 ζ153 ζ15−6 ζ156 ζ153 ζ15−3 ζ156 ζ15−6
105.1.3a1 C 3 0 0 3 3 3 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
105.1.3a2 C 3 0 0 3 3 3 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
105.1.3b1 C 3 0 0 3ζ35−14 3ζ3514 3ζ357 3ζ35−7 −ζ35−15−1−ζ355−ζ3510 ζ35−15+ζ355+ζ3510 0 0 0 0 0 0 0 0 −ζ35−14−ζ35−ζ3511−ζ3516 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 ζ35+ζ3511+ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516
105.1.3b2 C 3 0 0 3ζ3514 3ζ35−14 3ζ35−7 3ζ357 ζ35−15+ζ355+ζ3510 −ζ35−15−1−ζ355−ζ3510 0 0 0 0 0 0 0 0 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 −ζ35−14−ζ35−ζ3511−ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35+ζ3511+ζ3516 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358
105.1.3b3 C 3 0 0 3ζ35−14 3ζ3514 3ζ357 3ζ35−7 ζ35−15+ζ355+ζ3510 −ζ35−15−1−ζ355−ζ3510 0 0 0 0 0 0 0 0 ζ35+ζ3511+ζ3516 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 −ζ35−14−ζ35−ζ3511−ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516
105.1.3b4 C 3 0 0 3ζ3514 3ζ35−14 3ζ35−7 3ζ357 −ζ35−15−1−ζ355−ζ3510 ζ35−15+ζ355+ζ3510 0 0 0 0 0 0 0 0 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 ζ35+ζ3511+ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−ζ3511−ζ3516 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35−17+ζ35−12+ζ358
105.1.3b5 C 3 0 0 3ζ35−7 3ζ357 3ζ35−14 3ζ3514 −ζ35−15−1−ζ355−ζ3510 ζ35−15+ζ355+ζ3510 0 0 0 0 0 0 0 0 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−ζ3511−ζ3516 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 ζ35+ζ3511+ζ3516
105.1.3b6 C 3 0 0 3ζ357 3ζ35−7 3ζ3514 3ζ35−14 ζ35−15+ζ355+ζ3510 −ζ35−15−1−ζ355−ζ3510 0 0 0 0 0 0 0 0 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 −ζ35−14−ζ35−ζ3511−ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35+ζ3511+ζ3516 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515
105.1.3b7 C 3 0 0 3ζ35−7 3ζ357 3ζ35−14 3ζ3514 ζ35−15+ζ355+ζ3510 −ζ35−15−1−ζ355−ζ3510 0 0 0 0 0 0 0 0 ζ35−17+ζ35−12+ζ358 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35+ζ3511+ζ3516 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515 −ζ35−14−ζ35−ζ3511−ζ3516
105.1.3b8 C 3 0 0 3ζ357 3ζ35−7 3ζ3514 3ζ35−14 −ζ35−15−1−ζ355−ζ3510 ζ35−15+ζ355+ζ3510 0 0 0 0 0 0 0 0 ζ35−17−ζ35−13+ζ35−12+ζ35−10+ζ35−5+ζ354−ζ357+ζ359+ζ3511+ζ3516 ζ35−17+ζ35−12+ζ358 ζ35−13+ζ35−ζ354+ζ358−ζ359−ζ3514+ζ3515 ζ35+ζ3511+ζ3516 −ζ35−17−ζ35−12−ζ35−7−ζ358 −ζ35−17+ζ35−13−ζ35−12−ζ35−10−ζ35−5−ζ354−ζ359−ζ3511−ζ3516 −ζ35−14−ζ35−ζ3511−ζ3516 −ζ35−13−ζ35+ζ354−ζ358+ζ359−ζ3515

magma: CharacterTable(G);