metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊19D4, C10.692- (1+4), C4⋊D4⋊6D5, C5⋊3(Q8⋊5D4), C4.108(D4×D5), C4⋊C4.176D10, (D4×Dic5)⋊15C2, C20.224(C2×D4), C22⋊C4.5D10, D10⋊2Q8⋊19C2, Dic5⋊D4⋊9C2, (C2×D4).151D10, Dic5.44(C2×D4), C10.61(C22×D4), Dic5⋊3Q8⋊19C2, C20.17D4⋊14C2, C22⋊1(D4⋊2D5), C23.9(C22×D5), (C2×C10).142C24, (C2×C20).500C23, (C22×C4).218D10, Dic5.5D4⋊17C2, (C22×Dic10)⋊16C2, (D4×C10).116C22, C4⋊Dic5.204C22, (C22×C10).13C23, (C2×Dic5).65C23, (C4×Dic5).97C22, (C22×D5).61C23, C22.163(C23×D5), Dic5.14D4⋊16C2, C23.D5.20C22, D10⋊C4.11C22, (C22×C20).236C22, C10.D4.13C22, C2.27(D4.10D10), (C2×Dic10).253C22, (C22×Dic5).103C22, C2.34(C2×D4×D5), (C5×C4⋊D4)⋊7C2, (C4×C5⋊D4)⋊14C2, (C2×C10)⋊4(C4○D4), C10.80(C2×C4○D4), (C2×D4⋊2D5)⋊10C2, (C2×C4×D5).90C22, C2.31(C2×D4⋊2D5), (C5×C4⋊C4).138C22, (C2×C4).173(C22×D5), (C5×C22⋊C4).7C22, (C2×C5⋊D4).126C22, SmallGroup(320,1270)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 982 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×12], Q8 [×10], C23, C23 [×2], C23, D5, C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×5], C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×8], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×2], Q8⋊5D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5, D4⋊2D5 [×4], C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], Dic5.5D4 [×2], Dic5⋊3Q8, D10⋊2Q8, C4×C5⋊D4, D4×Dic5 [×2], C20.17D4, Dic5⋊D4 [×2], C5×C4⋊D4, C22×Dic10, C2×D4⋊2D5, Dic10⋊19D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- (1+4), C22×D5 [×7], Q8⋊5D4, D4×D5 [×2], D4⋊2D5 [×2], C23×D5, C2×D4×D5, C2×D4⋊2D5, D4.10D10, Dic10⋊19D4
Generators and relations
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 76 11 66)(2 75 12 65)(3 74 13 64)(4 73 14 63)(5 72 15 62)(6 71 16 61)(7 70 17 80)(8 69 18 79)(9 68 19 78)(10 67 20 77)(21 55 31 45)(22 54 32 44)(23 53 33 43)(24 52 34 42)(25 51 35 41)(26 50 36 60)(27 49 37 59)(28 48 38 58)(29 47 39 57)(30 46 40 56)(81 139 91 129)(82 138 92 128)(83 137 93 127)(84 136 94 126)(85 135 95 125)(86 134 96 124)(87 133 97 123)(88 132 98 122)(89 131 99 121)(90 130 100 140)(101 156 111 146)(102 155 112 145)(103 154 113 144)(104 153 114 143)(105 152 115 142)(106 151 116 141)(107 150 117 160)(108 149 118 159)(109 148 119 158)(110 147 120 157)
(1 44 132 112)(2 55 133 103)(3 46 134 114)(4 57 135 105)(5 48 136 116)(6 59 137 107)(7 50 138 118)(8 41 139 109)(9 52 140 120)(10 43 121 111)(11 54 122 102)(12 45 123 113)(13 56 124 104)(14 47 125 115)(15 58 126 106)(16 49 127 117)(17 60 128 108)(18 51 129 119)(19 42 130 110)(20 53 131 101)(21 87 144 65)(22 98 145 76)(23 89 146 67)(24 100 147 78)(25 91 148 69)(26 82 149 80)(27 93 150 71)(28 84 151 62)(29 95 152 73)(30 86 153 64)(31 97 154 75)(32 88 155 66)(33 99 156 77)(34 90 157 68)(35 81 158 79)(36 92 159 70)(37 83 160 61)(38 94 141 72)(39 85 142 63)(40 96 143 74)
(1 160)(2 141)(3 142)(4 143)(5 144)(6 145)(7 146)(8 147)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 136)(22 137)(23 138)(24 139)(25 140)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 100)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)
G:=sub<Sym(160)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,76,11,66)(2,75,12,65)(3,74,13,64)(4,73,14,63)(5,72,15,62)(6,71,16,61)(7,70,17,80)(8,69,18,79)(9,68,19,78)(10,67,20,77)(21,55,31,45)(22,54,32,44)(23,53,33,43)(24,52,34,42)(25,51,35,41)(26,50,36,60)(27,49,37,59)(28,48,38,58)(29,47,39,57)(30,46,40,56)(81,139,91,129)(82,138,92,128)(83,137,93,127)(84,136,94,126)(85,135,95,125)(86,134,96,124)(87,133,97,123)(88,132,98,122)(89,131,99,121)(90,130,100,140)(101,156,111,146)(102,155,112,145)(103,154,113,144)(104,153,114,143)(105,152,115,142)(106,151,116,141)(107,150,117,160)(108,149,118,159)(109,148,119,158)(110,147,120,157), (1,44,132,112)(2,55,133,103)(3,46,134,114)(4,57,135,105)(5,48,136,116)(6,59,137,107)(7,50,138,118)(8,41,139,109)(9,52,140,120)(10,43,121,111)(11,54,122,102)(12,45,123,113)(13,56,124,104)(14,47,125,115)(15,58,126,106)(16,49,127,117)(17,60,128,108)(18,51,129,119)(19,42,130,110)(20,53,131,101)(21,87,144,65)(22,98,145,76)(23,89,146,67)(24,100,147,78)(25,91,148,69)(26,82,149,80)(27,93,150,71)(28,84,151,62)(29,95,152,73)(30,86,153,64)(31,97,154,75)(32,88,155,66)(33,99,156,77)(34,90,157,68)(35,81,158,79)(36,92,159,70)(37,83,160,61)(38,94,141,72)(39,85,142,63)(40,96,143,74), (1,160)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,136)(22,137)(23,138)(24,139)(25,140)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,100)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111)>;
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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,76,11,66)(2,75,12,65)(3,74,13,64)(4,73,14,63)(5,72,15,62)(6,71,16,61)(7,70,17,80)(8,69,18,79)(9,68,19,78)(10,67,20,77)(21,55,31,45)(22,54,32,44)(23,53,33,43)(24,52,34,42)(25,51,35,41)(26,50,36,60)(27,49,37,59)(28,48,38,58)(29,47,39,57)(30,46,40,56)(81,139,91,129)(82,138,92,128)(83,137,93,127)(84,136,94,126)(85,135,95,125)(86,134,96,124)(87,133,97,123)(88,132,98,122)(89,131,99,121)(90,130,100,140)(101,156,111,146)(102,155,112,145)(103,154,113,144)(104,153,114,143)(105,152,115,142)(106,151,116,141)(107,150,117,160)(108,149,118,159)(109,148,119,158)(110,147,120,157), (1,44,132,112)(2,55,133,103)(3,46,134,114)(4,57,135,105)(5,48,136,116)(6,59,137,107)(7,50,138,118)(8,41,139,109)(9,52,140,120)(10,43,121,111)(11,54,122,102)(12,45,123,113)(13,56,124,104)(14,47,125,115)(15,58,126,106)(16,49,127,117)(17,60,128,108)(18,51,129,119)(19,42,130,110)(20,53,131,101)(21,87,144,65)(22,98,145,76)(23,89,146,67)(24,100,147,78)(25,91,148,69)(26,82,149,80)(27,93,150,71)(28,84,151,62)(29,95,152,73)(30,86,153,64)(31,97,154,75)(32,88,155,66)(33,99,156,77)(34,90,157,68)(35,81,158,79)(36,92,159,70)(37,83,160,61)(38,94,141,72)(39,85,142,63)(40,96,143,74), (1,160)(2,141)(3,142)(4,143)(5,144)(6,145)(7,146)(8,147)(9,148)(10,149)(11,150)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,136)(22,137)(23,138)(24,139)(25,140)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,100)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111) );
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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,76,11,66),(2,75,12,65),(3,74,13,64),(4,73,14,63),(5,72,15,62),(6,71,16,61),(7,70,17,80),(8,69,18,79),(9,68,19,78),(10,67,20,77),(21,55,31,45),(22,54,32,44),(23,53,33,43),(24,52,34,42),(25,51,35,41),(26,50,36,60),(27,49,37,59),(28,48,38,58),(29,47,39,57),(30,46,40,56),(81,139,91,129),(82,138,92,128),(83,137,93,127),(84,136,94,126),(85,135,95,125),(86,134,96,124),(87,133,97,123),(88,132,98,122),(89,131,99,121),(90,130,100,140),(101,156,111,146),(102,155,112,145),(103,154,113,144),(104,153,114,143),(105,152,115,142),(106,151,116,141),(107,150,117,160),(108,149,118,159),(109,148,119,158),(110,147,120,157)], [(1,44,132,112),(2,55,133,103),(3,46,134,114),(4,57,135,105),(5,48,136,116),(6,59,137,107),(7,50,138,118),(8,41,139,109),(9,52,140,120),(10,43,121,111),(11,54,122,102),(12,45,123,113),(13,56,124,104),(14,47,125,115),(15,58,126,106),(16,49,127,117),(17,60,128,108),(18,51,129,119),(19,42,130,110),(20,53,131,101),(21,87,144,65),(22,98,145,76),(23,89,146,67),(24,100,147,78),(25,91,148,69),(26,82,149,80),(27,93,150,71),(28,84,151,62),(29,95,152,73),(30,86,153,64),(31,97,154,75),(32,88,155,66),(33,99,156,77),(34,90,157,68),(35,81,158,79),(36,92,159,70),(37,83,160,61),(38,94,141,72),(39,85,142,63),(40,96,143,74)], [(1,160),(2,141),(3,142),(4,143),(5,144),(6,145),(7,146),(8,147),(9,148),(10,149),(11,150),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,136),(22,137),(23,138),(24,139),(25,140),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,127),(33,128),(34,129),(35,130),(36,131),(37,132),(38,133),(39,134),(40,135),(41,100),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 31 |
| 0 | 0 | 0 | 0 | 12 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 10 |
| 0 | 0 | 0 | 0 | 21 | 23 |
G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,12,0,0,0,0,31,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,21,0,0,0,0,10,23] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- (1+4) | D4×D5 | D4⋊2D5 | D4.10D10 |
| kernel | Dic10⋊19D4 | Dic5.14D4 | Dic5.5D4 | Dic5⋊3Q8 | D10⋊2Q8 | C4×C5⋊D4 | D4×Dic5 | C20.17D4 | Dic5⋊D4 | C5×C4⋊D4 | C22×Dic10 | C2×D4⋊2D5 | Dic10 | C4⋊D4 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes_{19}D_4 % in TeX
G:=Group("Dic10:19D4"); // GroupNames label
G:=SmallGroup(320,1270);
// by ID
G=gap.SmallGroup(320,1270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀