direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C28, C20⋊2C14, C140⋊6C2, Dic5⋊2C14, D10.2C14, C14.14D10, C70.19C22, C35⋊9(C2×C4), C5⋊2(C2×C28), C2.1(D5×C14), C10.2(C2×C14), (C7×Dic5)⋊5C2, (D5×C14).4C2, SmallGroup(280,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C28 |
Generators and relations for D5×C28
G = < a,b,c | a28=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C28
►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)
(1 132 64 99 33)(2 133 65 100 34)(3 134 66 101 35)(4 135 67 102 36)(5 136 68 103 37)(6 137 69 104 38)(7 138 70 105 39)(8 139 71 106 40)(9 140 72 107 41)(10 113 73 108 42)(11 114 74 109 43)(12 115 75 110 44)(13 116 76 111 45)(14 117 77 112 46)(15 118 78 85 47)(16 119 79 86 48)(17 120 80 87 49)(18 121 81 88 50)(19 122 82 89 51)(20 123 83 90 52)(21 124 84 91 53)(22 125 57 92 54)(23 126 58 93 55)(24 127 59 94 56)(25 128 60 95 29)(26 129 61 96 30)(27 130 62 97 31)(28 131 63 98 32)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 29)(26 30)(27 31)(28 32)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 113)(109 114)(110 115)(111 116)(112 117)
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), (1,132,64,99,33)(2,133,65,100,34)(3,134,66,101,35)(4,135,67,102,36)(5,136,68,103,37)(6,137,69,104,38)(7,138,70,105,39)(8,139,71,106,40)(9,140,72,107,41)(10,113,73,108,42)(11,114,74,109,43)(12,115,75,110,44)(13,116,76,111,45)(14,117,77,112,46)(15,118,78,85,47)(16,119,79,86,48)(17,120,80,87,49)(18,121,81,88,50)(19,122,82,89,51)(20,123,83,90,52)(21,124,84,91,53)(22,125,57,92,54)(23,126,58,93,55)(24,127,59,94,56)(25,128,60,95,29)(26,129,61,96,30)(27,130,62,97,31)(28,131,63,98,32), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,29)(26,30)(27,31)(28,32)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,113)(109,114)(110,115)(111,116)(112,117)>;
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), (1,132,64,99,33)(2,133,65,100,34)(3,134,66,101,35)(4,135,67,102,36)(5,136,68,103,37)(6,137,69,104,38)(7,138,70,105,39)(8,139,71,106,40)(9,140,72,107,41)(10,113,73,108,42)(11,114,74,109,43)(12,115,75,110,44)(13,116,76,111,45)(14,117,77,112,46)(15,118,78,85,47)(16,119,79,86,48)(17,120,80,87,49)(18,121,81,88,50)(19,122,82,89,51)(20,123,83,90,52)(21,124,84,91,53)(22,125,57,92,54)(23,126,58,93,55)(24,127,59,94,56)(25,128,60,95,29)(26,129,61,96,30)(27,130,62,97,31)(28,131,63,98,32), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,29)(26,30)(27,31)(28,32)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,113)(109,114)(110,115)(111,116)(112,117) );
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)], [(1,132,64,99,33),(2,133,65,100,34),(3,134,66,101,35),(4,135,67,102,36),(5,136,68,103,37),(6,137,69,104,38),(7,138,70,105,39),(8,139,71,106,40),(9,140,72,107,41),(10,113,73,108,42),(11,114,74,109,43),(12,115,75,110,44),(13,116,76,111,45),(14,117,77,112,46),(15,118,78,85,47),(16,119,79,86,48),(17,120,80,87,49),(18,121,81,88,50),(19,122,82,89,51),(20,123,83,90,52),(21,124,84,91,53),(22,125,57,92,54),(23,126,58,93,55),(24,127,59,94,56),(25,128,60,95,29),(26,129,61,96,30),(27,130,62,97,31),(28,131,63,98,32)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,29),(26,30),(27,31),(28,32),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,113),(109,114),(110,115),(111,116),(112,117)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 14G | ··· | 14R | 20A | 20B | 20C | 20D | 28A | ··· | 28L | 28M | ··· | 28X | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 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 | C7 | C14 | C14 | C14 | C28 | D5 | D10 | C4×D5 | C7×D5 | D5×C14 | D5×C28 |
| kernel | D5×C28 | C7×Dic5 | C140 | D5×C14 | C7×D5 | C4×D5 | Dic5 | C20 | D10 | D5 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of D5×C28 ►in GL2(𝔽29) generated by
| 3 | 0 |
| 0 | 3 |
| 0 | 25 |
| 22 | 5 |
| 24 | 20 |
| 22 | 5 |
G:=sub<GL(2,GF(29))| [3,0,0,3],[0,22,25,5],[24,22,20,5] >;
D5×C28 in GAP, Magma, Sage, TeX
D_5\times C_{28} % in TeX
G:=Group("D5xC28"); // GroupNames label
G:=SmallGroup(280,20);
// by ID
G=gap.SmallGroup(280,20);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-5,146,5604]);
// Polycyclic
G:=Group<a,b,c|a^28=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export