direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×D35, C35⋊3D7, C35⋊1C14, C72⋊1D5, C5⋊(C7×D7), C7⋊(C7×D5), (C7×C35)⋊2C2, SmallGroup(490,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C7×D35 |
Generators and relations for C7×D35
G = < a,b,c | a7=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D35
►On 70 points
Generators in S70
(1 31 26 21 16 11 6)(2 32 27 22 17 12 7)(3 33 28 23 18 13 8)(4 34 29 24 19 14 9)(5 35 30 25 20 15 10)(36 41 46 51 56 61 66)(37 42 47 52 57 62 67)(38 43 48 53 58 63 68)(39 44 49 54 59 64 69)(40 45 50 55 60 65 70)
(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)(36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70)
(1 36)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)
G:=sub<Sym(70)| (1,31,26,21,16,11,6)(2,32,27,22,17,12,7)(3,33,28,23,18,13,8)(4,34,29,24,19,14,9)(5,35,30,25,20,15,10)(36,41,46,51,56,61,66)(37,42,47,52,57,62,67)(38,43,48,53,58,63,68)(39,44,49,54,59,64,69)(40,45,50,55,60,65,70), (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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,36)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)>;
G:=Group( (1,31,26,21,16,11,6)(2,32,27,22,17,12,7)(3,33,28,23,18,13,8)(4,34,29,24,19,14,9)(5,35,30,25,20,15,10)(36,41,46,51,56,61,66)(37,42,47,52,57,62,67)(38,43,48,53,58,63,68)(39,44,49,54,59,64,69)(40,45,50,55,60,65,70), (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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,36)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,60)(13,59)(14,58)(15,57)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37) );
G=PermutationGroup([[(1,31,26,21,16,11,6),(2,32,27,22,17,12,7),(3,33,28,23,18,13,8),(4,34,29,24,19,14,9),(5,35,30,25,20,15,10),(36,41,46,51,56,61,66),(37,42,47,52,57,62,67),(38,43,48,53,58,63,68),(39,44,49,54,59,64,69),(40,45,50,55,60,65,70)], [(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),(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70)], [(1,36),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,60),(13,59),(14,58),(15,57),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37)]])
133 conjugacy classes
| class | 1 | 2 | 5A | 5B | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 35A | ··· | 35CR |
| order | 1 | 2 | 5 | 5 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 35 | ··· | 35 |
| size | 1 | 35 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 35 | ··· | 35 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | D5 | D7 | C7×D5 | D35 | C7×D7 | C7×D35 |
| kernel | C7×D35 | C7×C35 | D35 | C35 | C72 | C35 | C7 | C7 | C5 | C1 |
| # reps | 1 | 1 | 6 | 6 | 2 | 3 | 12 | 12 | 18 | 72 |
Matrix representation of C7×D35 ►in GL2(𝔽71) generated by
| 20 | 0 |
| 0 | 20 |
| 10 | 0 |
| 56 | 64 |
| 9 | 67 |
| 20 | 62 |
G:=sub<GL(2,GF(71))| [20,0,0,20],[10,56,0,64],[9,20,67,62] >;
C7×D35 in GAP, Magma, Sage, TeX
C_7\times D_{35} % in TeX
G:=Group("C7xD35"); // GroupNames label
G:=SmallGroup(490,8);
// by ID
G=gap.SmallGroup(490,8);
# by ID
G:=PCGroup([4,-2,-7,-5,-7,674,6723]);
// Polycyclic
G:=Group<a,b,c|a^7=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export