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Properties

Label 35T12
Degree $35$
Order $280$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(35, 12);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7\times F_5$
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:  (1,21)(2,25)(3,24)(4,23)(5,22)(6,16)(7,20)(8,19)(9,18)(10,17)(12,15)(13,14)(26,31)(27,35)(28,34)(29,33)(30,32), (1,12,5,14)(2,15,4,11)(3,13)(6,7,10,9)(16,32,20,34)(17,35,19,31)(18,33)(21,27,25,29)(22,30,24,26)(23,28)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$14$:  $D_{7}$
$20$:  $F_5$
$28$:  $D_{14}$
$40$:  $F_{5}\times C_2$
$56$:  28T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Degree 7: $D_{7}$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $280=2^{3} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  280.32
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A 7A1 7A2 7A3 10A 14A1 14A3 14A5 28A1 28A-1 28A3 28A-3 28A5 28A-5 35A1 35A2 35A3
Size 1 5 7 35 5 5 35 35 4 2 2 2 28 10 10 10 10 10 10 10 10 10 8 8 8
2 P 1A 1A 1A 1A 2A 2A 2A 2A 5A 7A2 7A3 7A1 5A 7A3 7A2 7A1 14A1 14A1 14A3 14A3 14A5 14A5 35A3 35A1 35A2
5 P 1A 2A 2B 2C 4A-1 4A1 4B-1 4B1 5A 7A3 7A1 7A2 10A 14A5 14A1 14A3 28A3 28A-3 28A5 28A-5 28A-1 28A1 35A1 35A2 35A3
7 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 7A2 7A3 7A1 2B 14A1 14A3 14A5 28A5 28A-5 28A-1 28A1 28A-3 28A3 7A1 7A2 7A3
Type
280.32.1a 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
280.32.1b 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
280.32.1c 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
280.32.1d 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
280.32.1e1 C 1 −1 −1 1 −i i i −i 1 1 1 1 −1 −1 −1 −1 i −i −i i i −i 1 1 1
280.32.1e2 C 1 −1 −1 1 i −i −i i 1 1 1 1 −1 −1 −1 −1 −i i i −i −i i 1 1 1
280.32.1f1 C 1 −1 1 −1 −i i −i i 1 1 1 1 1 −1 −1 −1 i −i −i i i −i 1 1 1
280.32.1f2 C 1 −1 1 −1 i −i i −i 1 1 1 1 1 −1 −1 −1 −i i i −i −i i 1 1 1
280.32.2a1 R 2 2 0 0 2 2 0 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
280.32.2a2 R 2 2 0 0 2 2 0 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
280.32.2a3 R 2 2 0 0 2 2 0 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
280.32.2b1 R 2 2 0 0 −2 −2 0 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
280.32.2b2 R 2 2 0 0 −2 −2 0 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
280.32.2b3 R 2 2 0 0 −2 −2 0 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
280.32.2c1 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 0 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 ζ283+ζ2811 −ζ283−ζ2811 ζ285+ζ289 −ζ285−ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283−ζ285+ζ287−ζ289+ζ2811 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282
280.32.2c2 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 0 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 −ζ283−ζ2811 ζ283+ζ2811 −ζ285−ζ289 ζ285+ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282
280.32.2c3 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 0 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 −ζ285−ζ289 ζ285+ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283+ζ2811 −ζ283−ζ2811 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286
280.32.2c4 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 0 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 ζ285+ζ289 −ζ285−ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283−ζ2811 ζ283+ζ2811 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286
280.32.2c5 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 0 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283−ζ2811 ζ283+ζ2811 −ζ285−ζ289 ζ285+ζ289 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284
280.32.2c6 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 0 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283+ζ2811 −ζ283−ζ2811 ζ285+ζ289 −ζ285−ζ289 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284
280.32.4a R 4 0 4 0 0 0 0 0 −1 4 4 4 −1 0 0 0 0 0 0 0 0 0 −1 −1 −1
280.32.4b R 4 0 −4 0 0 0 0 0 −1 4 4 4 1 0 0 0 0 0 0 0 0 0 −1 −1 −1
280.32.8a1 R 8 0 0 0 0 0 0 0 −2 4ζ7−3+4ζ73 4ζ7−1+4ζ7 4ζ7−2+4ζ72 0 0 0 0 0 0 0 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
280.32.8a2 R 8 0 0 0 0 0 0 0 −2 4ζ7−2+4ζ72 4ζ7−3+4ζ73 4ζ7−1+4ζ7 0 0 0 0 0 0 0 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
280.32.8a3 R 8 0 0 0 0 0 0 0 −2 4ζ7−1+4ζ7 4ζ7−2+4ζ72 4ζ7−3+4ζ73 0 0 0 0 0 0 0 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7

magma: CharacterTable(G);