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

Group invariants

Abstract group:  $D_7\times F_5$
magma: IdentifyGroup(G);
 
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
magma: NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{14},1^{7}$ $5$ $2$ $14$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,15)(12,14)(16,20)(17,19)(21,25)(22,24)(26,30)(27,29)(31,35)(32,34)$
2B $2^{15},1^{5}$ $7$ $2$ $15$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30)$
2C $2^{17},1$ $35$ $2$ $17$ $( 1,15)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)$
4A1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 1, 4, 5, 2)( 6, 9,10, 7)(11,14,15,12)(16,19,20,17)(21,24,25,22)(26,29,30,27)(31,34,35,32)$
4A-1 $4^{7},1^{7}$ $5$ $4$ $21$ $( 1, 2, 5, 4)( 6, 7,10, 9)(11,12,15,14)(16,17,20,19)(21,22,25,24)(26,27,30,29)(31,32,35,34)$
4B1 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1,14, 5,12)( 2,11, 4,15)( 3,13)( 6, 9,10, 7)(16,34,20,32)(17,31,19,35)(18,33)(21,29,25,27)(22,26,24,30)(23,28)$
4B-1 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1,12, 5,14)( 2,15, 4,11)( 3,13)( 6, 7,10, 9)(16,32,20,34)(17,35,19,31)(18,33)(21,27,25,29)(22,30,24,26)(23,28)$
5A $5^{7}$ $4$ $5$ $28$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23)( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
7A2 $7^{5}$ $2$ $7$ $30$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33)( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
7A3 $7^{5}$ $2$ $7$ $30$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28)( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
10A $10^{3},5$ $28$ $10$ $31$ $( 1,33, 5,32, 4,31, 3,35, 2,34)( 6,28,10,27, 9,26, 8,30, 7,29)(11,23,15,22,14,21,13,25,12,24)(16,18,20,17,19)$
14A1 $14^{2},7$ $10$ $14$ $32$ $( 1,15,21,35, 6,20,26, 5,11,25,31,10,16,30)( 2,14,22,34, 7,19,27, 4,12,24,32, 9,17,29)( 3,13,23,33, 8,18,28)$
14A3 $14^{2},7$ $10$ $14$ $32$ $( 1,10,11,20,21,30,31, 5, 6,15,16,25,26,35)( 2, 9,12,19,22,29,32, 4, 7,14,17,24,27,34)( 3, 8,13,18,23,28,33)$
14A5 $14^{2},7$ $10$ $14$ $32$ $( 1,20,31,15,26,10,21, 5,16,35,11,30, 6,25)( 2,19,32,14,27, 9,22, 4,17,34,12,29, 7,24)( 3,18,33,13,28, 8,23)$
28A1 $28,7$ $10$ $28$ $33$ $( 1,12,25,34, 6,17,30, 4,11,22,35, 9,16,27, 5,14,21,32,10,19,26, 2,15,24,31, 7,20,29)( 3,13,23,33, 8,18,28)$
28A-1 $28,7$ $10$ $28$ $33$ $( 1, 9,15,17,21,29,35, 2, 6,14,20,22,26,34, 5, 7,11,19,25,27,31, 4,10,12,16,24,30,32)( 3, 8,13,18,23,28,33)$
28A3 $28,7$ $10$ $28$ $33$ $( 1,19,35,12,26, 9,25, 2,16,34,15,27, 6,24, 5,17,31,14,30, 7,21, 4,20,32,11,29,10,22)( 3,18,33,13,28, 8,23)$
28A-3 $28,7$ $10$ $28$ $33$ $( 1, 7,15,19,21,27,35, 4, 6,12,20,24,26,32, 5, 9,11,17,25,29,31, 2,10,14,16,22,30,34)( 3, 8,13,18,23,28,33)$
28A5 $28,7$ $10$ $28$ $33$ $( 1,17,35,14,26, 7,25, 4,16,32,15,29, 6,22, 5,19,31,12,30, 9,21, 2,20,34,11,27,10,24)( 3,18,33,13,28, 8,23)$
28A-5 $28,7$ $10$ $28$ $33$ $( 1,14,25,32, 6,19,30, 2,11,24,35, 7,16,29, 5,12,21,34,10,17,26, 4,15,22,31, 9,20,27)( 3,13,23,33, 8,18,28)$
35A1 $35$ $8$ $35$ $34$ $( 1,33,30,22,19,11, 8, 5,32,29,21,18,15, 7, 4,31,28,25,17,14, 6, 3,35,27,24,16,13,10, 2,34,26,23,20,12, 9)$
35A2 $35$ $8$ $35$ $34$ $( 1,28,20, 7,34,21,13, 5,27,19, 6,33,25,12, 4,26,18,10,32,24,11, 3,30,17, 9,31,23,15, 2,29,16, 8,35,22,14)$
35A3 $35$ $8$ $35$ $34$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$

Malle's constant $a(G)$:     $1/14$

magma: ConjugacyClasses(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 7A1 7A2 7A3 5A 7A3 7A2 7A1 14A5 14A1 14A3 14A1 14A3 14A5 35A3 35A1 35A2
5 P 1A 2A 2B 2C 4A-1 4A1 4B-1 4B1 5A 7A2 7A3 7A1 10A 14A3 14A5 14A1 28A-1 28A-3 28A5 28A3 28A-5 28A1 35A1 35A2 35A3
7 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 7A1 7A2 7A3 2B 14A5 14A1 14A3 28A-3 28A-5 28A-1 28A5 28A1 28A3 7A2 7A3 7A1
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);
 

Regular extensions

Data not computed