direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C35, C5⋊C70, C35⋊3C10, C52⋊1C14, (C5×C35)⋊4C2, SmallGroup(350,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C35 |
Generators and relations for D5×C35
G = < a,b,c | a35=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×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 8 15 22 29)(2 9 16 23 30)(3 10 17 24 31)(4 11 18 25 32)(5 12 19 26 33)(6 13 20 27 34)(7 14 21 28 35)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)
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), (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)>;
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), (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62) );
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)], [(1,8,15,22,29),(2,9,16,23,30),(3,10,17,24,31),(4,11,18,25,32),(5,12,19,26,33),(6,13,20,27,34),(7,14,21,28,35),(36,64,57,50,43),(37,65,58,51,44),(38,66,59,52,45),(39,67,60,53,46),(40,68,61,54,47),(41,69,62,55,48),(42,70,63,56,49)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62)]])
140 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 35A | ··· | 35X | 35Y | ··· | 35CF | 70A | ··· | 70X |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
140 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 | D5 | C5×D5 | C7×D5 | D5×C35 |
| kernel | D5×C35 | C5×C35 | C7×D5 | C5×D5 | C35 | C52 | D5 | C5 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 4 | 6 | 4 | 6 | 24 | 24 | 2 | 8 | 12 | 48 |
Matrix representation of D5×C35 ►in GL2(𝔽71) generated by
| 12 | 0 |
| 0 | 12 |
| 25 | 0 |
| 55 | 54 |
| 17 | 18 |
| 55 | 54 |
G:=sub<GL(2,GF(71))| [12,0,0,12],[25,55,0,54],[17,55,18,54] >;
D5×C35 in GAP, Magma, Sage, TeX
D_5\times C_{35} % in TeX
G:=Group("D5xC35"); // GroupNames label
G:=SmallGroup(350,6);
// by ID
G=gap.SmallGroup(350,6);
# by ID
G:=PCGroup([4,-2,-5,-7,-5,4483]);
// Polycyclic
G:=Group<a,b,c|a^35=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export