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Properties

Label 35T12
35T12 1 12 1->12 21 1->21 2 15 2->15 25 2->25 3 13 3->13 24 3->24 4 11 4->11 23 4->23 5 14 5->14 22 5->22 6 7 6->7 16 6->16 10 7->10 20 7->20 8 19 8->19 9 9->6 18 9->18 10->9 17 10->17 11->2 12->5 12->15 13->14 14->1 15->4 32 16->32 35 17->35 33 18->33 31 19->31 34 20->34 27 21->27 30 22->30 28 23->28 26 24->26 29 25->29 26->22 26->31 27->25 27->35 28->34 29->21 29->33 30->24 30->32 31->17 32->20 34->16 35->19
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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Copy content magma:G := TransitiveGroup(35, 12);
 

Group invariants

Abstract group:  $D_7\times F_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $280=2^{3} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $35$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,21)(2,25)(3,24)(4,23)(5,22)(6,16)(7,20)(8,19)(9,18)(10,17)(12,15)(13,14)(26,31)(27,35)(28,34)(29,33)(30,32)$, $(1,12,5,14)(2,15,4,11)(3,13)(6,7,10,9)(16,32,20,34)(17,35,19,31)(18,33)(21,27,25,29)(22,30,24,26)(23,28)$
Copy content 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 $2^{15},1^{5}$ $7$ $2$ $15$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)$
2C $2^{17},1$ $35$ $2$ $17$ $( 1,33)( 2,32)( 3,31)( 4,35)( 5,34)( 6,28)( 7,27)( 8,26)( 9,30)(10,29)(11,23)(12,22)(13,21)(14,25)(15,24)(16,18)(19,20)$
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 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1,30, 3,29)( 2,27)( 4,26, 5,28)( 6,25, 8,24)( 7,22)( 9,21,10,23)(11,20,13,19)(12,17)(14,16,15,18)(31,35,33,34)$
4B-1 $4^{7},2^{3},1$ $35$ $4$ $24$ $( 1,29, 3,30)( 2,27)( 4,28, 5,26)( 6,24, 8,25)( 7,22)( 9,23,10,21)(11,19,13,20)(12,17)(14,18,15,16)(31,34,33,35)$
5A $5^{7}$ $4$ $5$ $28$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
7A1 $7^{5}$ $2$ $7$ $30$ $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18)( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$
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 $7^{5}$ $2$ $7$ $30$ $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13)( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$
10A $10^{3},5$ $28$ $10$ $31$ $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,30, 9,28, 7,26,10,29, 8,27)(11,25,14,23,12,21,15,24,13,22)(16,20,19,18,17)$
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 $14^{2},7$ $10$ $14$ $32$ $( 1,35,26,25,16,15, 6, 5,31,30,21,20,11,10)( 2,34,27,24,17,14, 7, 4,32,29,22,19,12, 9)( 3,33,28,23,18,13, 8)$
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 $28,7$ $10$ $28$ $33$ $( 1, 7,15,19,21,27,35, 4, 6,12,20,24,26,32, 5, 9,11,17,25,29,31, 2,10,14,16,22,30,34)( 3, 8,13,18,23,28,33)$
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 $28,7$ $10$ $28$ $33$ $( 1,16,31,11,26, 6,21)( 2,18,35,14,27, 8,25, 4,17,33,15,29, 7,23, 5,19,32,13,30, 9,22, 3,20,34,12,28,10,24)$
28A-3 $28,7$ $10$ $28$ $33$ $( 1,19,33,15,26, 9,23, 5,16,34,13,30, 6,24, 3,20,31,14,28,10,21, 4,18,35,11,29, 8,25)( 2,17,32,12,27, 7,22)$
28A5 $28,7$ $10$ $28$ $33$ $( 1,12,25,34, 6,17,30, 4,11,22,35, 9,16,27, 5,14,21,32,10,19,26, 2,15,24,31, 7,20,29)( 3,13,23,33, 8,18,28)$
28A-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 $35$ $8$ $35$ $34$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14,26, 8,25, 2,19,31,13,30, 7,24)$
35A2 $35$ $8$ $35$ $34$ $( 1,35,29,23,17,11,10, 4,33,27,21,20,14, 8, 2,31,30,24,18,12, 6, 5,34,28,22,16,15, 9, 3,32,26,25,19,13, 7)$
35A3 $35$ $8$ $35$ $34$ $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$

Malle's constant $a(G)$:     $1/14$

Copy content 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 7A2 7A3 7A1 5A 7A1 7A3 7A2 14A1 14A1 14A3 14A3 14A5 14A5 35A2 35A3 35A1
5 P 1A 2A 2B 2C 4A-1 4A1 4B-1 4B1 5A 7A3 7A1 7A2 10A 14A3 14A5 14A1 28A3 28A-3 28A5 28A-5 28A-1 28A1 35A3 35A1 35A2
7 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 7A2 7A3 7A1 2B 14A5 14A1 14A3 28A5 28A-5 28A-1 28A1 28A-3 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 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
280.32.2a2 R 2 2 0 0 2 2 0 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
280.32.2a3 R 2 2 0 0 2 2 0 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
280.32.2b1 R 2 2 0 0 −2 −2 0 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
280.32.2b2 R 2 2 0 0 −2 −2 0 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
280.32.2b3 R 2 2 0 0 −2 −2 0 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
280.32.2c1 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 0 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 ζ283+ζ2811 ζ285+ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ285−ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 −ζ283−ζ2811 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286
280.32.2c2 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 0 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 −ζ283−ζ2811 −ζ285−ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ285+ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 ζ283+ζ2811 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286
280.32.2c3 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 0 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 −ζ285−ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283−ζ2811 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283+ζ2811 ζ285+ζ289 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284
280.32.2c4 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 0 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 ζ285+ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ283+ζ2811 ζ283−ζ285+ζ287−ζ289+ζ2811 −ζ283−ζ2811 −ζ285−ζ289 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284
280.32.2c5 C 2 −2 0 0 −2ζ287 2ζ287 0 0 2 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 0 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 −ζ283+ζ285−ζ287+ζ289−ζ2811 −ζ283−ζ2811 ζ285+ζ289 ζ283+ζ2811 −ζ285−ζ289 ζ283−ζ285+ζ287−ζ289+ζ2811 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282
280.32.2c6 C 2 −2 0 0 2ζ287 −2ζ287 0 0 2 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 0 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 ζ283−ζ285+ζ287−ζ289+ζ2811 ζ283+ζ2811 −ζ285−ζ289 −ζ283−ζ2811 ζ285+ζ289 −ζ283+ζ285−ζ287+ζ289−ζ2811 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282
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 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
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 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7
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 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73

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Regular extensions

Data not computed