direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D7×C35, C7⋊C70, C35⋊2C14, C72⋊1C10, (C7×C35)⋊3C2, SmallGroup(490,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C35 |
Generators and relations for D7×C35
G = < a,b,c | a35=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C35
►On 70 points
Generators in S70
(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 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)(36 61 51 41 66 56 46)(37 62 52 42 67 57 47)(38 63 53 43 68 58 48)(39 64 54 44 69 59 49)(40 65 55 45 70 60 50)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)
G:=sub<Sym(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), (1,11,21,31,6,16,26)(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)(36,61,51,41,66,56,46)(37,62,52,42,67,57,47)(38,63,53,43,68,58,48)(39,64,54,44,69,59,49)(40,65,55,45,70,60,50), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)>;
G:=Group( (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), (1,11,21,31,6,16,26)(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)(36,61,51,41,66,56,46)(37,62,52,42,67,57,47)(38,63,53,43,68,58,48)(39,64,54,44,69,59,49)(40,65,55,45,70,60,50), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50) );
G=PermutationGroup([[(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)], [(1,11,21,31,6,16,26),(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),(36,61,51,41,66,56,46),(37,62,52,42,67,57,47),(38,63,53,43,68,58,48),(39,64,54,44,69,59,49),(40,65,55,45,70,60,50)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50)]])
175 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 7G | ··· | 7AA | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 35A | ··· | 35X | 35Y | ··· | 35DD | 70A | ··· | 70X |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 7 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | 7 | 7 | 7 | 7 | ··· | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 | D7 | C5×D7 | C7×D7 | D7×C35 |
| kernel | D7×C35 | C7×C35 | C7×D7 | C5×D7 | C72 | C35 | D7 | C7 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 4 | 6 | 4 | 6 | 24 | 24 | 3 | 12 | 18 | 72 |
Matrix representation of D7×C35 ►in GL3(𝔽71) generated by
| 16 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 1 | 0 | 0 |
| 0 | 45 | 26 |
| 0 | 0 | 30 |
| 70 | 0 | 0 |
| 0 | 10 | 20 |
| 0 | 27 | 61 |
G:=sub<GL(3,GF(71))| [16,0,0,0,48,0,0,0,48],[1,0,0,0,45,0,0,26,30],[70,0,0,0,10,27,0,20,61] >;
D7×C35 in GAP, Magma, Sage, TeX
D_7\times C_{35} % in TeX
G:=Group("D7xC35"); // GroupNames label
G:=SmallGroup(490,5);
// by ID
G=gap.SmallGroup(490,5);
# by ID
G:=PCGroup([4,-2,-5,-7,-7,6723]);
// Polycyclic
G:=Group<a,b,c|a^35=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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