direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C20, C28⋊2C10, C140⋊5C2, D14.C10, Dic7⋊2C10, C10.14D14, C70.14C22, C7⋊1(C2×C20), C35⋊8(C2×C4), C2.1(C10×D7), C14.2(C2×C10), (C5×Dic7)⋊5C2, (C10×D7).2C2, SmallGroup(280,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C20 |
Generators and relations for D7×C20
G = < a,b,c | a20=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C20
►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 57 121 104 37 95 68)(2 58 122 105 38 96 69)(3 59 123 106 39 97 70)(4 60 124 107 40 98 71)(5 41 125 108 21 99 72)(6 42 126 109 22 100 73)(7 43 127 110 23 81 74)(8 44 128 111 24 82 75)(9 45 129 112 25 83 76)(10 46 130 113 26 84 77)(11 47 131 114 27 85 78)(12 48 132 115 28 86 79)(13 49 133 116 29 87 80)(14 50 134 117 30 88 61)(15 51 135 118 31 89 62)(16 52 136 119 32 90 63)(17 53 137 120 33 91 64)(18 54 138 101 34 92 65)(19 55 139 102 35 93 66)(20 56 140 103 36 94 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 121)(38 122)(39 123)(40 124)(41 99)(42 100)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)
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), (1,57,121,104,37,95,68)(2,58,122,105,38,96,69)(3,59,123,106,39,97,70)(4,60,124,107,40,98,71)(5,41,125,108,21,99,72)(6,42,126,109,22,100,73)(7,43,127,110,23,81,74)(8,44,128,111,24,82,75)(9,45,129,112,25,83,76)(10,46,130,113,26,84,77)(11,47,131,114,27,85,78)(12,48,132,115,28,86,79)(13,49,133,116,29,87,80)(14,50,134,117,30,88,61)(15,51,135,118,31,89,62)(16,52,136,119,32,90,63)(17,53,137,120,33,91,64)(18,54,138,101,34,92,65)(19,55,139,102,35,93,66)(20,56,140,103,36,94,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(41,99)(42,100)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)>;
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), (1,57,121,104,37,95,68)(2,58,122,105,38,96,69)(3,59,123,106,39,97,70)(4,60,124,107,40,98,71)(5,41,125,108,21,99,72)(6,42,126,109,22,100,73)(7,43,127,110,23,81,74)(8,44,128,111,24,82,75)(9,45,129,112,25,83,76)(10,46,130,113,26,84,77)(11,47,131,114,27,85,78)(12,48,132,115,28,86,79)(13,49,133,116,29,87,80)(14,50,134,117,30,88,61)(15,51,135,118,31,89,62)(16,52,136,119,32,90,63)(17,53,137,120,33,91,64)(18,54,138,101,34,92,65)(19,55,139,102,35,93,66)(20,56,140,103,36,94,67), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(41,99)(42,100)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98) );
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)], [(1,57,121,104,37,95,68),(2,58,122,105,38,96,69),(3,59,123,106,39,97,70),(4,60,124,107,40,98,71),(5,41,125,108,21,99,72),(6,42,126,109,22,100,73),(7,43,127,110,23,81,74),(8,44,128,111,24,82,75),(9,45,129,112,25,83,76),(10,46,130,113,26,84,77),(11,47,131,114,27,85,78),(12,48,132,115,28,86,79),(13,49,133,116,29,87,80),(14,50,134,117,30,88,61),(15,51,135,118,31,89,62),(16,52,136,119,32,90,63),(17,53,137,120,33,91,64),(18,54,138,101,34,92,65),(19,55,139,102,35,93,66),(20,56,140,103,36,94,67)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,121),(38,122),(39,123),(40,124),(41,99),(42,100),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 14A | 14B | 14C | 20A | ··· | 20H | 20I | ··· | 20P | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 14 | 14 | 14 | 20 | ··· | 20 | 20 | ··· | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D7 | D14 | C4×D7 | C5×D7 | C10×D7 | D7×C20 |
| kernel | D7×C20 | C5×Dic7 | C140 | C10×D7 | C5×D7 | C4×D7 | Dic7 | C28 | D14 | D7 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 3 | 3 | 6 | 12 | 12 | 24 |
Matrix representation of D7×C20 ►in GL2(𝔽41) generated by
| 8 | 0 |
| 0 | 8 |
| 40 | 20 |
| 32 | 15 |
| 26 | 34 |
| 32 | 15 |
G:=sub<GL(2,GF(41))| [8,0,0,8],[40,32,20,15],[26,32,34,15] >;
D7×C20 in GAP, Magma, Sage, TeX
D_7\times C_{20} % in TeX
G:=Group("D7xC20"); // GroupNames label
G:=SmallGroup(280,15);
// by ID
G=gap.SmallGroup(280,15);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-7,106,6004]);
// Polycyclic
G:=Group<a,b,c|a^20=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export