direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×D35, C35⋊2D5, C35⋊1C10, C52⋊1D7, C7⋊(C5×D5), C5⋊(C5×D7), (C5×C35)⋊2C2, SmallGroup(350,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C5×D35 |
Generators and relations for C5×D35
G = < a,b,c | a5=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D35
►On 70 points
Generators in S70
(1 15 29 8 22)(2 16 30 9 23)(3 17 31 10 24)(4 18 32 11 25)(5 19 33 12 26)(6 20 34 13 27)(7 21 35 14 28)(36 57 43 64 50)(37 58 44 65 51)(38 59 45 66 52)(39 60 46 67 53)(40 61 47 68 54)(41 62 48 69 55)(42 63 49 70 56)
(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 38)(2 37)(3 36)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)
G:=sub<Sym(70)| (1,15,29,8,22)(2,16,30,9,23)(3,17,31,10,24)(4,18,32,11,25)(5,19,33,12,26)(6,20,34,13,27)(7,21,35,14,28)(36,57,43,64,50)(37,58,44,65,51)(38,59,45,66,52)(39,60,46,67,53)(40,61,47,68,54)(41,62,48,69,55)(42,63,49,70,56), (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), (1,38)(2,37)(3,36)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)>;
G:=Group( (1,15,29,8,22)(2,16,30,9,23)(3,17,31,10,24)(4,18,32,11,25)(5,19,33,12,26)(6,20,34,13,27)(7,21,35,14,28)(36,57,43,64,50)(37,58,44,65,51)(38,59,45,66,52)(39,60,46,67,53)(40,61,47,68,54)(41,62,48,69,55)(42,63,49,70,56), (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), (1,38)(2,37)(3,36)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39) );
G=PermutationGroup([[(1,15,29,8,22),(2,16,30,9,23),(3,17,31,10,24),(4,18,32,11,25),(5,19,33,12,26),(6,20,34,13,27),(7,21,35,14,28),(36,57,43,64,50),(37,58,44,65,51),(38,59,45,66,52),(39,60,46,67,53),(40,61,47,68,54),(41,62,48,69,55),(42,63,49,70,56)], [(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)], [(1,38),(2,37),(3,36),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39)]])
95 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 35A | ··· | 35BT |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 35 | ··· | 35 |
| size | 1 | 35 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 35 | 35 | 35 | 35 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | D5 | D7 | C5×D5 | C5×D7 | D35 | C5×D35 |
| kernel | C5×D35 | C5×C35 | D35 | C35 | C35 | C52 | C7 | C5 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 2 | 3 | 8 | 12 | 12 | 48 |
Matrix representation of C5×D35 ►in GL2(𝔽71) generated by
| 54 | 0 |
| 0 | 54 |
| 60 | 0 |
| 65 | 58 |
| 13 | 28 |
| 65 | 58 |
G:=sub<GL(2,GF(71))| [54,0,0,54],[60,65,0,58],[13,65,28,58] >;
C5×D35 in GAP, Magma, Sage, TeX
C_5\times D_{35} % in TeX
G:=Group("C5xD35"); // GroupNames label
G:=SmallGroup(350,7);
// by ID
G=gap.SmallGroup(350,7);
# by ID
G:=PCGroup([4,-2,-5,-5,-7,482,4803]);
// Polycyclic
G:=Group<a,b,c|a^5=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export