metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D70.C2, D35⋊2C4, Dic7⋊2D5, Dic5⋊2D7, C14.3D10, C10.3D14, C70.3C22, C7⋊1(C4×D5), C5⋊2(C4×D7), C35⋊6(C2×C4), C2.3(D5×D7), (C7×Dic5)⋊2C2, (C5×Dic7)⋊2C2, SmallGroup(280,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D70.C2 |
Generators and relations for D70.C2
G = < a,b,c | a70=b2=1, c2=a35, bab=a-1, cac-1=a41, cbc-1=a40b >
Smallest permutation representation of D70.C2
►On 140 points
Generators in S140
(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)(71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)
(1 112 36 77)(2 83 37 118)(3 124 38 89)(4 95 39 130)(5 136 40 101)(6 107 41 72)(7 78 42 113)(8 119 43 84)(9 90 44 125)(10 131 45 96)(11 102 46 137)(12 73 47 108)(13 114 48 79)(14 85 49 120)(15 126 50 91)(16 97 51 132)(17 138 52 103)(18 109 53 74)(19 80 54 115)(20 121 55 86)(21 92 56 127)(22 133 57 98)(23 104 58 139)(24 75 59 110)(25 116 60 81)(26 87 61 122)(27 128 62 93)(28 99 63 134)(29 140 64 105)(30 111 65 76)(31 82 66 117)(32 123 67 88)(33 94 68 129)(34 135 69 100)(35 106 70 71)
G:=sub<Sym(140)| (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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,112,36,77)(2,83,37,118)(3,124,38,89)(4,95,39,130)(5,136,40,101)(6,107,41,72)(7,78,42,113)(8,119,43,84)(9,90,44,125)(10,131,45,96)(11,102,46,137)(12,73,47,108)(13,114,48,79)(14,85,49,120)(15,126,50,91)(16,97,51,132)(17,138,52,103)(18,109,53,74)(19,80,54,115)(20,121,55,86)(21,92,56,127)(22,133,57,98)(23,104,58,139)(24,75,59,110)(25,116,60,81)(26,87,61,122)(27,128,62,93)(28,99,63,134)(29,140,64,105)(30,111,65,76)(31,82,66,117)(32,123,67,88)(33,94,68,129)(34,135,69,100)(35,106,70,71)>;
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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112), (1,112,36,77)(2,83,37,118)(3,124,38,89)(4,95,39,130)(5,136,40,101)(6,107,41,72)(7,78,42,113)(8,119,43,84)(9,90,44,125)(10,131,45,96)(11,102,46,137)(12,73,47,108)(13,114,48,79)(14,85,49,120)(15,126,50,91)(16,97,51,132)(17,138,52,103)(18,109,53,74)(19,80,54,115)(20,121,55,86)(21,92,56,127)(22,133,57,98)(23,104,58,139)(24,75,59,110)(25,116,60,81)(26,87,61,122)(27,128,62,93)(28,99,63,134)(29,140,64,105)(30,111,65,76)(31,82,66,117)(32,123,67,88)(33,94,68,129)(34,135,69,100)(35,106,70,71) );
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),(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(101,122),(102,121),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112)], [(1,112,36,77),(2,83,37,118),(3,124,38,89),(4,95,39,130),(5,136,40,101),(6,107,41,72),(7,78,42,113),(8,119,43,84),(9,90,44,125),(10,131,45,96),(11,102,46,137),(12,73,47,108),(13,114,48,79),(14,85,49,120),(15,126,50,91),(16,97,51,132),(17,138,52,103),(18,109,53,74),(19,80,54,115),(20,121,55,86),(21,92,56,127),(22,133,57,98),(23,104,58,139),(24,75,59,110),(25,116,60,81),(26,87,61,122),(27,128,62,93),(28,99,63,134),(29,140,64,105),(30,111,65,76),(31,82,66,117),(32,123,67,88),(33,94,68,129),(34,135,69,100),(35,106,70,71)]])
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 35 | 35 | 5 | 5 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D7 | D10 | D14 | C4×D5 | C4×D7 | D5×D7 | D70.C2 |
| kernel | D70.C2 | C7×Dic5 | C5×Dic7 | D70 | D35 | Dic7 | Dic5 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of D70.C2 ►in GL4(𝔽281) generated by
| 280 | 37 | 0 | 0 |
| 244 | 244 | 0 | 0 |
| 0 | 0 | 227 | 7 |
| 0 | 0 | 267 | 7 |
| 280 | 37 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 41 | 280 |
| 0 | 0 | 275 | 240 |
| 280 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 |
| 0 | 0 | 138 | 15 |
| 0 | 0 | 229 | 143 |
G:=sub<GL(4,GF(281))| [280,244,0,0,37,244,0,0,0,0,227,267,0,0,7,7],[280,0,0,0,37,1,0,0,0,0,41,275,0,0,280,240],[280,0,0,0,0,280,0,0,0,0,138,229,0,0,15,143] >;
D70.C2 in GAP, Magma, Sage, TeX
D_{70}.C_2 % in TeX
G:=Group("D70.C2"); // GroupNames label
G:=SmallGroup(280,9);
// by ID
G=gap.SmallGroup(280,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,26,328,6004]);
// Polycyclic
G:=Group<a,b,c|a^70=b^2=1,c^2=a^35,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^40*b>;
// generators/relations
Export