↑

Properties

Label 35T22
Degree $35$
Order $1260$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_7:\GL(2,4)$

Downloads

Learn more

Show commands: Magma

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

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:\GL(2,4)$
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:  (1,15,24,34,7,19,28,2,13,21,35,6,20,30)(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)
magma: Generators(G);
 

Low degree resolvents

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1260=2^{2} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1260.61
magma: IdentifyGroup(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-1 15A-2 15A1 21A1 21A-1 35A-1 35A2 35A1 35A-2
3 P 1A 2A 1A 1A 1A 1A 1A 5A2 5A1 2A 2A 7A-1 7A1 14A-1 14A1 5A2 5A1 5A2 5A1 7A-1 7A1 35A1 35A-2 35A-1 35A2
5 P 1A 2A 3A-1 3A1 3B 3C-1 3C1 1A 1A 6A-1 6A1 7A-1 7A1 14A-1 14A1 3A1 3A1 3A-1 3A-1 21A-1 21A1 7A1 7A-1 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 15A1 15A2 15A-1 3B 3B 5A2 5A1 5A2 5A1
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 ζ3−1 ζ3−1 ζ3 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 ζ3 ζ3−1 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 −ζ7−3−1−ζ7−ζ72 ζ7−3+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+ζ7+ζ72 −ζ7−3−1−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 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
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 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
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+ζ154−ζ155 −ζ15−ζ154 −1+ζ15+ζ152−ζ153+ζ154+ζ157 0 0 −ζ15−6−ζ156 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−3−ζ153
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 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −ζ15−ζ154 0 0 −ζ15−6−ζ156 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−3−ζ153
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 −ζ15−ζ154 −1+ζ15+ζ154−ζ155 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 0 0 −ζ15−3−ζ153 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−6−ζ156
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+ζ152−ζ153+ζ154+ζ157 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ154−ζ155 0 0 −ζ15−3−ζ153 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−6−ζ156
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 −ζ3−1 −ζ3−1 −ζ3 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 −ζ3 −ζ3−1 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 ζ35−13−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
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 ζ35−14−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
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 −ζ35−13+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
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 −ζ35−14+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
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+1+ζ7+ζ72 −ζ7−3−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−ζ7−ζ72 ζ7−3+1+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

magma: CharacterTable(G);